Michael line and Morita’s conjectures

Posted on September 9, 2018 by Dan Ma

This post discusses Michael line from the point of view of the three conjectures of Kiiti Morita.

K. Morita defined the notion of P-spaces in [7]. The definition of P-spaces is discussed here in considerable details. K. Morita also proved that a space $X$ is a normal P-space if and only if the product $X \times Y$ is normal for every metrizable space $Y$ . As a result of this characterization, the notion of normal P-space (a space that is a normal space and a P-space) is useful in the study of products of normal spaces. Just to be clear, we say a space is a non-normal P-space (i.e. a space that is not a normal P-space) if the space is a normal space that is not a P-space.

K. Morita formulated his three conjectures in 1976. The statements of the conjectures are given below. Here is a basic discussion of the three conjectures. The notion of normal P-spaces is a theme that runs through the three conjectures. The conjectures are actually theorems since 2001 [2].

Here’s where Michael line comes into the discussion. Based on the characterization of normal P-spaces mentioned above, to find a normal space that is not a P-space (a non-normal P-space), we would need to find a non-normal product $X \times Y$ such that one of the factors is a metric space and the other factor is a normal space. The first such example in ZFC is from an article by E. Michael in 1963 (found here and here). In this example, the normal space is $M$ , which came be known as the Michael line, and the metric space is $\mathbb{P}$ , the space of irrational numbers (as a subspace of the real line). Their product $M \times \mathbb{P}$ is not normal. A basic discussion of the Michael line is found here.

Because $M \times \mathbb{P}$ is not normal, the Michael line $M$ is not a normal P-space. Prior to E. Michael’s 1963 article, we have to reach back to 1955 to find an example of a non-normal product where one factor is a metric space. In 1955, M. E. Rudin used a Souslin line to construct a Dowker space, which is a normal space whose product with the closed unit interval is not normal. The existence of a Souslin line was shown to be independent of ZFC in the late 1960s. In 1971, Rudin constructed a Dowker space in ZFC. Thus finding a normal space that is not a normal P-space (finding a non-normal product $X \times Y$ where one factor is a metric space and the other factor is a normal space) is not a trivial matter.

Morita’s Three Conjectures

We show that the Michael line illustrates perfectly the three conjectures of K. Morita. Here’s the statements.

Morita’s Conjecture I. Let $X$ be a space. If the product $X \times Y$ is normal for every normal space $Y$ then $X$ is a discrete space.

Morita’s Conjecture II. Let $X$ be a space. If the product $X \times Y$ is normal for every normal P-space $Y$ then $X$ is a metrizable space.

Morita’s Conjecture III. Let $X$ be a space. If the product $X \times Y$ is normal for every normal countably paracompact space $Y$ then $X$ is a metrizable $\sigma$ -locally compact space.

The contrapositive statement of Morita’s conjecture I is that for any non-discrete space $X$ , there exists a normal space $Y$ such that $X \times Y$ is not normal. Thus any non-discrete space is paired with a normal space for forming a non-normal product. The Michael line $M$ is paired with the space of irrational numbers $\mathbb{P}$ . Obviously, the space $\mathbb{P}$ is paired with the Michael line $M$ .

The contrapositive statement of Morita’s conjecture II is that for any non-metrizable space $X$ , there exists a normal P-space $Y$ such that $X \times Y$ is not normal. The pairing is more specific than for conjecture I. Any non-metrizable space is paired with a normal P-space to form a non-normal product. As illustration, the Michael line $M$ is not metrizable. The space $\mathbb{P}$ of irrational numbers is a metric space and hence a normal P-space. Here, $M$ is paired with $\mathbb{P}$ to form a non-normal product.

The contrapositive statement of Morita’s conjecture III is that for any space $X$ that is not both metrizable and $\sigma$ -locally compact, there exists a normal countably paracompact space $Y$ such that $X \times Y$ is not normal. Note that the space $\mathbb{P}$ is not $\sigma$ -locally compact (see Theorem 4 here). The Michael line $M$ is paracompact and hence normal and countably paracompact. Thus the metric non- $\sigma$ -locally compact $\mathbb{P}$ is paired with normal countably paracompact $M$ to form a non-normal product. Here, the metric space $\mathbb{P}$ is paired with the non-normal P-space $M$ .

In each conjecture, each space in a certain class of spaces is paired with one space in another class to form a non-normal product. For Morita’s conjecture I, each non-discrete space is paired with a normal space. For conjecture II, each non-metrizable space is paired with a normal P-space. For conjecture III, each metrizable but non- $\sigma$ -locally compact is paired with a normal countably paracompact space to form a non-normal product. Note that the paired normal countably paracompact space would be a non-normal P-space.

Michael line as an example of a non-normal P-space is a great tool to help us walk through the three conjectures of Morita. Are there other examples of non-normal P-spaces? Dowker spaces mentioned above (normal spaces whose products with the closed unit interval are not normal) are non-normal P-spaces. Note that conjecture II guarantees a normal P-space to match every non-metric space for forming a non-normal product. Conjecture III guarantees a non-normal P-space to match every metrizable non- $\sigma$ -locally compact space for forming a non-normal product. Based on the conjectures, examples of normal P-spaces and non-normal P-spaces, though may be hard to find, are guaranteed to exist.

We give more examples below to further illustrate the pairings for conjecture II and conjecture III. As indicated above, non-normal P-spaces are hard to come by. Some of the examples below are constructed using additional axioms beyond ZFC. The additional examples still give an impression that the availability of non-normal P-spaces, though guaranteed to exist, is limited.

Examples of Normal P-Spaces

One example is based on this classic theorem: for any normal space $X$ , $X$ is paracompact if and only if the product $X \times \beta X$ is normal. Here $\beta X$ is the Stone-Cech compactification of the completely regular space $X$ . Thus any normal but not paracompact space $X$ (a non-metrizable space) is paired with $\beta X$ , a normal P-space, to form a non-normal product.

Naturally, the next class of non-metrizable spaces to be discussed should be the paracompact spaces that are not metrizable. If there is a readily available theorem to provide a normal P-space for each non-metrizable paracompact space, then there would be a simple proof of Morita’s conjecture II. The eventual solution of conjecture II is far from simple [2]. We narrow the focus to the non-metrizable compact spaces.

Consider this well known result: for any infinite compact space $X$ , the product $\omega_1 \times X$ is normal if and only if the space $X$ has countable tightness (see Theorem 1 here). Thus any compact space with uncountable tightness is paired with $\omega_1$ , the space of all countable ordinals, to form a non-normal product. The space $\omega_1$ , being a countably compact space, is a normal P-space. A proof that normal countably compact space is a normal P-space is given here.

We now handle the case for non-metrizable compact spaces with countable tightness. In this case, compactness is not needed. For spaces with countable tightness, consider this result: every space with countable tightness, whose products with all perfectly normal spaces are normal, must be metrizable [3] (see Corollary 7). Thus any non-metrizable space with countable tightness is paired with some perfectly normal space to form a non-normal product. Any reader interested in what these perfectly normal spaces are can consult [3]. Note that perfectly normal spaces are normal P-spaces (see here for a proof).

Examples of Non-Normal P-Spaces

Another non-normal product is $X_B \times B$ where $B \subset \mathbb{R}$ is a Bernstein set and $X_B$ is the space with the real line as the underlying set such that points in $B$ are isolated and points in $\mathbb{R}-B$ retain the usual open sets. The set $B \subset \mathbb{R}$ is said to be a Bernstein set if every uncountable closed subset of the real line contains a point in B and contains a point in the complement of B. Such a set can be constructed using transfinite induction as shown here. The product $X_B \times B$ is not normal where $B$ is considered a subspace of the real line. The proof is essentially the same proof that shows $M \times \mathbb{P}$ is not normal (see here). The space $X_B$ is a Lindelof space. It is not a normal P-space since its product with $B$ , a separable metric space, is not normal. However, this example is essentially the same example as the Michael line since the same technique and proof are used. On the one hand, the $X_B \times B$ example seems like an improvement over Michael line example since the first factor $X_B$ is Lindelof. On the other hand, it is inferior than the Michael line example since the second factor $B$ is not completely metrizable.

Moving away from the idea of Michael, there exist a Lindelof space and a completely metrizable (but not separable) space whose product is of weight $\omega_1$ and is not normal [5]. This would be a Lindelof space that is a non-normal P-space. However, this example is not as elementary as the Michael line, making it not as effective as an illustration of Morita’s three conjectures.

The next set of non-normal P-spaces requires set theory. A Michael space is a Lindelof space whose product with $\mathbb{P}$ , the space of irrational numbers, is not normal. Michael problem is the question: is there a Michael space in ZFC? It is known that a Michael space can be constructed using continuum hypothesis [6] or using Martin’s axiom [1]. The construction using continuum hypothesis has been discussed in this blog (see here). The question of whether there exists a Michael space in ZFC is still unsolved.

The existence of a Michael space is equivalent to the existence of a Lindelof space and a separable completely metrizable space whose product is non-normal [4]. A Michael space, in the context of the discussion in this post, is a non-normal P-space.

The discussion in this post shows that the example of the Michael line and other examples of non-normal P-spaces are useful tools to illustrate Morita’s three conjectures.

Reference

Alster K.,On the product of a Lindelof space and the space of irrationals under Martin’s Axiom, Proc. Amer. Math. Soc., Vol. 110, 543-547, 1990.
Balogh Z.,Normality of product spaces and Morita’s conjectures, Topology Appl., Vol. 115, 333-341, 2001.
Chiba K., Przymusinski T., Rudin M. E.Nonshrinking open covers and K. Morita’s duality conjectures, Topology Appl., Vol. 22, 19-32, 1986.
Lawrence L. B., The influence of a small cardinal on the product of a Lindelof space and the irrationals, Proc. Amer. Math. Soc., 110, 535-542, 1990.
Lawrence L. B., A ZFC Example (of Minimum Weight) of a Lindelof Space and a Completely Metrizable Space with a Nonnormal Product, Proc. Amer. Math. Soc., 124, No 2, 627-632, 1996.
Michael E., Paracompactness and the Lindelof property in nite and countable cartesian products, Compositio Math., 23, 199-214, 1971.
Morita K., Products of Normal Spaces with Metric Spaces, Math. Ann., Vol. 154, 365-382, 1964.
Rudin M. E., A Normal Space $X$ for which $X \times I$ is not Normal, Fund. Math., 73, 179-186, 1971.

$\text{ }$

Dan Ma math

Daniel Ma mathematics

$\copyright$ 2018 – Dan Ma

Three conjectures of K Morita

Posted on August 28, 2018 by Dan Ma

This post discusses the three conjectures that were proposed by K. Morita in 1976. These conjectures concern normality in product spaces. To start the discussion, here’s the conjectures.

Morita’s Conjecture I. Let $X$ be a space. The product $X \times Y$ is normal for every normal space $Y$ if and only if $X$ is a discrete space.

Morita’s Conjecture II. Let $X$ be a space. The product $X \times Y$ is normal for every normal P-space $Y$ if and only if $X$ is a metrizable space.

Morita’s Conjecture III. Let $X$ be a space. The product $X \times Y$ is normal for every normal countably paracompact space $Y$ if and only if $X$ is a metrizable $\sigma$ -locally compact space.

These statements are no longer conjectures. Partial results appeared after the conjectures were proposed in 1976. The complete resolution of the conjectures came in 2001 in a paper by Zoli Balogh [5]. Though it is more appropriate to call these statements theorems, it is still convenient to call them conjectures. Just know that they are now known results rather open problems to be solved. The focus here is not on the evolution of the solutions. Instead, we discuss the relations among the three conjectures and why they are amazing results in the study of normality in product spaces.

As discussed below, in each of these conjectures, one direction is true based on prior known theorems (see Theorem 1, Theorem 2 and Theorem 4 below). The conjectures can be stated as follows.

Morita’s Conjecture I. Let $X$ be a space. If the product $X \times Y$ is normal for every normal space $Y$ then $X$ is a discrete space.

Morita’s Conjecture II. Let $X$ be a space. If the product $X \times Y$ is normal for every normal P-space $Y$ then $X$ is a metrizable space.

Morita’s Conjecture III. Let $X$ be a space. If the product $X \times Y$ is normal for every normal countably paracompact space $Y$ then $X$ is a metrizable $\sigma$ -locally compact space.

P-spaces are defined by K. Morita [11]. He proved that a space $X$ is a normal P-space if and only if the product $X \times Y$ is normal for every metrizable space $Y$ (see theorem 2 below). Normal P-spaces are also discussed here. A space $X$ is $\sigma$ -locally compact space if $X$ is the union of countably many locally compact subspaces each of which is also closed subspace of $X$ .

As we will see below, these conjectures are also called duality conjectures because they are duals of known results.

[2] is a survey of Morita’s conjecture.

Duality Conjectures

Here’s three theorems that are duals to the conjectures.

Theorem 1
Let $X$ be a space. The product space $X \times Y$ is normal for every discrete space $Y$ if and only if $X$ is normal.

Theorem 2
Let $X$ be a space. The product space $X \times Y$ is normal for every metrizable space $Y$ if and only if $X$ is a normal P-space.

Theorem 3
Let $X$ be a space. The product space $X \times Y$ is normal for every metrizable $\sigma$ -locally compact space $Y$ if and only if $X$ is normal countably paracompact.

The key words in red are for emphasis. In each of these three theorems, if we switch the two key words in red, we would obtain the statements for the conjectures. In this sense, the conjectures are called duality conjectures since they are duals of known results.

Theorem 1 is actually not found in the literature. It is an easy theorem. Theorem 2, found in [11], is a characterization of normal P-space (discussed here). Theorem 3 is a well known result based on the following theorem by K. Morita [10].

Theorem 4
Let $Y$ be a metrizable space. Then the product $X \times Y$ is normal for every normal countably paracompact space $X$ if and only if $Y$ is a $\sigma$ -locally compact space.

We now show that Theorem 3 can be established using Theorem 4. Theorem 4 is also Theorem 3.5 in p. 111 of [2]. A proof of Theorem 4 is found in Theorem 1.8 in p. 130 of [8].

Proof of Theorem 3
$\Longleftarrow$ Suppose $X$ is normal and countably paracompact. Let $Y$ be a metrizable $\sigma$ -locally compact space. By Theorem 4, $X \times Y$ is normal.

$\Longrightarrow$ This direction uses Dowker’s theorem. We give a contrapositive proof. Suppose that $X$ is not both normal and countably paracompact. Case 1. $X$ is not normal. Then $X \times \{ y \}$ is not normal where $\{ y \}$ is any one-point discrete space. Case 2. $X$ is normal and not countably paracompact. This means that $X$ is a Dowker space. Then $X \times [0,1]$ is not normal. In either case, $X \times Y$ is not normal for some compact metric space. Thus $X \times Y$ is not normal for some $\sigma$ -locally compact metric space. This completes the proof of Theorem 3. $\square$

The First and Third Conjectures

The first conjecture of Morita was proved by Atsuji [1] and Rudin [13] in 1978. The proof in [13] is a constructive proof. The key to that solution is to define a $\kappa$ -Dowker space. Suppose $X$ is a non-discrete space. Let $\kappa$ be the least cardinal of a non-discrete subspace of $X$ . Then construct a $\kappa$ -Dowker space $Y$ as in [13]. It follows that $X \times Y$ is not normal. The proof that $X \times Y$ is not normal is discussed here.

Conjecture III was confirmed by Balogh in 1998 [4]. We show here that the first and third conjectures of Morita can be confirmed by assuming the second conjecture.

Conjecture II implies Conjecture I
We give a contrapositive proof of Conjecture I. Suppose that $X$ is not discrete. We wish to find a normal space $Y$ such that $X \times Y$ is not normal. Consider two cases for $X$ . Case 1. $X$ is not metrizable. By Conjecture II, $X \times Y$ is not normal for some normal P-space $Y$ . Case 2. $X$ is metrizable. Since $X$ is infinite and metric, $X$ would contain an infinite compact metric space $S$ . For example, $X$ contains a non-trivial convergent sequence and let $S$ be a convergence sequence plus the limit point. Let $Y$ be a Dowker space. Then the product $S \times Y$ is not normal. It follows that $X \times Y$ is not normal. Thus there exists a normal space $Y$ such that $X \times Y$ is not normal in either case. $\square$

Conjecture II implies Conjecture III
Suppose that the product $X \times Y$ is normal for every normal and countably paracompact space $Y$ . Since any normal P-space is a normal countably paracompact space, $X \times Y$ is normal for every normal and P-space $Y$ . By Conjecture II, $X$ is metrizable. By Theorem 4, $X$ is $\sigma$ -locally compact. $\square$

The Second Conjecture

The above discussion shows that a complete solution to the three conjectures hinges on the resolution of the second conjecture. A partial resolution came in 1986 [6]. In that paper, it was shown that under V = L, conjecture II is true.

The complete solution of the second conjecture is given in a paper of Balogh [5] in 2001. The path to Balogh’s proof is through a conjecture of M. E. Rudin identified as Conjecture 9.

Rudin’s Conjecture 9. There exists a normal P-space $X$ such that some uncountable increasing open cover of $X$ cannot be shrunk.

Conjecture 9 was part of a set of 14 conjectures stated in [14]. It is also discussed in [7]. In [6], conjecture 9 was shown to be equivalent to Morita’s second conjecture. In [5], Balogh used his technique for constructing a Dowker space of cardinality continuum to obtain a space as described in conjecture 9.

The resolution of conjecture II is considered to be one of Balogh greatest hits [3].

Abundance of Non-Normal Products

One immediate observation from Morita’s conjecture I is that existence of non-normal products is wide spread. Conjecture I indicates that every normal non-discrete space $X$ is paired with some normal space $Y$ such that their product is not normal. So every normal non-discrete space forms a non-normal product with some normal space. Given any normal non-discrete space (no matter how nice it is or how exotic it is), it can always be paired with another normal space (sometimes paired with itself) for a non-normal product.

Suppose we narrow the focus to spaces that are normal and non-metrizable. Then any such space $X$ is paired with some normal P-space $Y$ to form a non-normal product space (Morita’s conjecture II). By narrowing the focus on $X$ to the non-metrizable spaces, we obtain more clarity on the paired space to form non-normal product, namely a normal P-space. As an example, let $X$ be the Michael line (normal and non-metrizable). It is well known that $X$ in this case is paired with $\mathbb{P}$ , the space of irrational numbers with the usual Euclidean topology, to form a non-normal product (discussed here).

Another example is $X$ being the Sorgenfrey line. It is well known that $X$ in this case is paired with itself to form a non-normal product (discussed here). Morita’s conjectures are powerful indication that these two non-normal products are not isolated phenomena.

Another interesting observation about conjecture II is that normal P-spaces are not productive with respect to normality. More specifically, for any non-metrizable normal P-space $X$ , conjecture II tells us that there exists another normal P-space $Y$ such that $X \times Y$ is not normal.

Now we narrow the focus to spaces that are metrizable but not $\sigma$ -locally compact. For any such space $X$ , conjecture III tells us that $X$ is paired with a normal countably paracompact space $Y$ to form a non-normal product. Using the Michael line example, this time let $X=\mathbb{P}$ , the space of irrational numbers, which is a metric space that is not $\sigma$ -locally compact. The paired normal and countably paracompact space $Y$ is the Michael line.

Each conjecture is about existence of a normal $Y$ that is paired with a given $X$ to form a non-normal product. For Conjecture I, the given $X$ is from a wide class (normal non-discrete). As a result, there is not much specific information on the paired $Y$ , other than that it is normal. For Conjectures II and III, the given space $X$ is from narrower classes. As a result, there is more information on the paired $Y$ .

The concept of Dowker spaces runs through the three conjectures, especially the first conjecture. Dowker spaces and $\kappa$ -Dowker spaces provide reliable pairing for non-normal products. In fact this is one way to prove conjecture I [13], also see here. For any normal space $X$ with a countable non-discrete subspace, the product of $X$ and any Dowker space is not normal (discussed here). For any normal space $X$ such that the least cardinality of a non-discrete subspace is an uncountable cardinal $\kappa$ , the product $X \times Y$ is not normal where $Y$ is a $\kappa$ -Dowker space as constructed in [13], also discussed here.

In finding a normal pair $Y$ for a normal space $X$ , if we do not care about $Y$ having a high degree of normal productiveness (e.g. normal P or normal countably paracompact), we can always let $Y$ be a Dowker space or $\kappa$ -Dowker space. In fact, if the starting space $X$ is a metric space, the normal pair for a non-normal product (by definition) has to be a Dowker space. For example, if $X=[0,1]$ , then the normal space $Y$ such that $X \times Y$ is by definition a Dowker space. The search for a Dowker space spanned a period of 20 years. For the real line $\mathbb{R}$ , the normal pair for a non-normal product is also a Dowker space. For “nice” spaces such as metric spaces, finding a normal space to form non-normal product is no trivial problem.

Reference

Atsuji M.,On normality of the product of two spaces, General Topology and Its Relation to Modern Analysis and Algebra (Proc. Fourth Prague Topology sympos., 1976), Part B, 25–27, 1977.
Atsuji M.,Normality of product spaces I, in: K. Morita, J. Nagata (Eds.), Topics in General
Topology, North-Holland, Amsterdam, 81–116, 1989.
Burke D., Gruenhage G.,Zoli, Top. Proc., Vol. 27, No 1, i-xxii, 2003.
Balogh Z.,Normality of product spaces and K. Morita’s third conjecture, Topology Appl., Vol. 84, 185-198, 1998.
Balogh Z.,Normality of product spaces and Morita’s conjectures, Topology Appl., Vol. 115, 333-341, 2001.
Chiba K., Przymusinski T., Rudin M. E.Nonshrinking open covers and K. Morita’s duality conjectures, Topology Appl., Vol. 22, 19-32, 1986.
Gruenhage G.,Mary Ellen’s Conjectures,, Special Issue honoring the memory of Mary Ellen Rudin, Topology Appl., Vol. 195, 15-25, 2015.
Hoshina T.,Normality of product spaces II, in: K. Morita, J. Nagata (Eds.), Topics in General Topology, North-Holland, Amsterdam, 121–158, 1989.
Morita K., On the Product of a Normal Space with a Metric Space, Proc. Japan Acad., Vol. 39, 148-150, 1963. (article information; paper)
Morita K., Products of Normal Spaces with Metric Spaces II, Sci. Rep. Tokyo Kyoiku Dagaiku Sec A, 8, 87-92, 1963.
Morita K., Products of Normal Spaces with Metric Spaces, Math. Ann., Vol. 154, 365-382, 1964.
Morita K., Nagata J., Topics in General Topology, Elsevier Science Publishers, B. V., The Netherlands, 1989.
Rudin M. E., $\kappa$ -Dowker Spaces, Czechoslovak Mathematical Journal, 28, No.2, 324-326, 1978.
Rudin M. E., Some conjectures, in: Open Problems in Topology, J. van Mill and G.M. Reed,
eds., North Holland, 184–193, 1990.
Telgárski R., A characterization of P-spaces, Proc. Japan Acad., Vol. 51, 802–807, 1975.

$\text{ }$

Dan Ma math

Daniel Ma mathematics

$\copyright$ 2018 – Dan Ma

Morita’s normal P-space

Posted on August 18, 2018 by Dan Ma

In this post we discuss K. Morita’s notion of P-space, which is a useful and interesting concept in the study of normality of product spaces.

The Definition

In [1] and [2], Morita defined the notion of P-spaces. First some notations. Let $\kappa$ be a cardinal number such that $\kappa \ge 1$ . Conveniently, $\kappa$ is identified by the set of all ordinals preceding $\kappa$ . Let $\Gamma$ be the set of all finite sequences $(\alpha_1,\alpha_2,\cdots,\alpha_n)$ where $n=1,2,\cdots$ and all $\alpha_i < \kappa$ . Let $X$ be a space. The collection $\left\{A_\sigma \subset X: \sigma \in \Gamma \right\}$ is said to be decreasing if this condition holds: for any $\sigma \in \Gamma$ and $\delta \in \Gamma$ with

$\sigma =(\alpha_1,\alpha_2,\cdots,\alpha_n)$

$\delta =(\beta_1,\beta_2,\cdots,\beta_n, \cdots, \beta_m)$

such that $n<m$ and such that $\alpha_i=\beta_i$ for all $i \le n$ , we have $A_{\delta} \subset A_{\sigma}$ . On the other hand, the collection $\left\{A_\sigma \subset X: \sigma \in \Gamma \right\}$ is said to be increasing if for any $\sigma \in \Gamma$ and $\delta \in \Gamma$ as described above, we have $A_{\sigma} \subset A_{\delta}$ .

The space $X$ is a P-space if for any cardinal $\kappa \ge 1$ and for any decreasing collection $\left\{F_\sigma \subset X: \sigma \in \Gamma \right\}$ of closed subsets of $X$ , there exists open set $U_\sigma$ for each $\sigma \in \Gamma$ with $F_\sigma \subset U_\sigma$ such that for any countably infinite sequence $(\alpha_1,\alpha_2,\cdots,\alpha_n,\cdots)$ where each finite subsequence $\sigma_n=(\alpha_1,\alpha_2,\cdots,\alpha_n)$ is an element of $\Gamma$ , if $\bigcap_{n=1}^\infty F_{\sigma_n}=\varnothing$ , then $\bigcap_{n=1}^\infty U_{\sigma_n}=\varnothing$ .

By switching closed sets and open sets and by switching decreasing collection and increasing collection, the following is an alternative but equivalent definition of P-spaces.

The space $X$ is a P-space if for any cardinal $\kappa \ge 1$ and for any increasing collection $\left\{U_\sigma \subset X: \sigma \in \Gamma \right\}$ of open subsets of $X$ , there exists closed set $F_\sigma$ for each $\sigma \in \Gamma$ with $F_\sigma \subset U_\sigma$ such that for any countably infinite sequence $(\alpha_1,\alpha_2,\cdots,\alpha_n,\cdots)$ where each finite subsequence $\sigma_n=(\alpha_1,\alpha_2,\cdots,\alpha_n)$ is an element of $\Gamma$ , if $\bigcup_{n=1}^\infty U_{\sigma_n}=X$ , then $\bigcup_{n=1}^\infty F_{\sigma_n}=X$ .

Note that the definition is per cardinal number $\kappa \ge 1$ . To bring out more precision, we say a space $X$ is a P( $\kappa$ )-space of it satisfies the definition for P-space for the cardinal $\kappa$ . Of course if a space is a P( $\kappa$ )-space for all $\kappa \ge 1$ , then it is a P-space.

There is also a game characterization of P-spaces [4].

A Specific Case

It is instructive to examine a specific case of the definition. Let $\kappa=1=\{ 0 \}$ . In other words, let’s look what what a P(1)-space looks like. The elements of the index set $\Gamma$ are simply finite sequences of 0’s. The relevant information about an element of $\Gamma$ is its length (i.e. a positive integer). Thus the closed sets $F_\sigma$ in the definition are essentially indexed by integers. For the case of $\kappa=1$ , the definition can be stated as follows:

For any decreasing sequence $F_1 \supset F_2 \supset F_3 \cdots$ of closed subsets of $X$ , there exist $U_1,U_2,U_3,\cdots$ , open subsets of $X$ , such that $F_n \subset U_n$ for all $n$ and such that if $\bigcap_{n=1}^\infty F_n=\varnothing$ then $\bigcap_{n=1}^\infty U_n=\varnothing$ .

The above condition implies the following condition.

For any decreasing sequence $F_1 \supset F_2 \supset F_3 \cdots$ of closed subsets of $X$ such that $\bigcap_{n=1}^\infty F_n=\varnothing$ , there exist $U_1,U_2,U_3,\cdots$ , open subsets of $X$ , such that $F_n \subset U_n$ for all $n$ and such that $\bigcap_{n=1}^\infty U_n=\varnothing$ .

The last condition is one of the conditions in Dowker’s Theorem (condition 6 in Theorem 1 in this post and condition 7 in Theorem 1 in this post). Recall that Dowker’s theorem states that a normal space $X$ is countably paracompact if and only if the last condition holds if and only of the product $X \times Y$ is normal for every infinite compact metric space $Y$ . Thus if a normal space $X$ is a P(1)-space, it is countably paracompact. More importantly P(1) space is about normality in product spaces where one factor is a class of metric spaces, namely the compact metric spaces.

Based on the above discussion, any normal space $X$ that is a P-space is a normal countably paracompact space.

The definition for P(1)-space is identical to one combinatorial condition in Dowker’s theorem which says that any decreasing sequence of closed sets with empty intersection has an open expansion that also has empty intersection.

For P( $\kappa$ )-space where $\kappa>1$ , the decreasing family of closed sets are no longer indexed by the integers. Instead the decreasing closed sets are indexed by finite sequences of elements of $\kappa$ . The index set $\Gamma$ would be more like a tree structure. However the look and feel of P-space is like the combinatorial condition in Dowker’s theorem. The decreasing closed sets are expanded by open sets. For any “path in the tree” (an infinite sequence of elements of $\kappa$ ), if the closed sets along the path has empty intersection, then the corresponding open sets would have empty intersection.

Not surprisingly, the notion of P-spaces is about normality in product spaces where one factor is a metric space. In fact, this is precisely the characterization of P-spaces (see Theorem 1 and Theorem 2 below).

A Characterization of P-Space

Morita gave the following characterization of P-spaces among normal spaces. The following theorems are found in [2].

Theorem 1
Let $X$ be a space. The space $X$ is a normal P-space if and only if the product space $X \times Y$ is normal for every metrizable space $Y$ .

Thus the combinatorial definition involving decreasing families of closed sets being expanded by open sets is equivalent to a statement that is much easier to understand. A space that is normal and a P-space is precisely a normal space that is productively normal with every metric space. The following theorem is Theorem 1 broken out for each cardinal $\kappa$ .

Theorem 2
Let $X$ be a space and let $\kappa \ge \omega$ . Then $X$ is a normal P( $\kappa$ )-space if and only if the product space $X \times Y$ is normal for every metric space $Y$ of weight $\kappa$ .

Theorem 2 only covers the infinite cardinals $\kappa$ starting with the countably infinite cardinal. Where are the P( $n$ )-spaces placed where $n$ are the positive integers? The following theorem gives the answer.

Theorem 3
Let $X$ be a space. Then $X$ is a normal P(2)-space if and only if the product space $X \times Y$ is normal for every separable metric space $Y$ .

According to Theorem 2, $X$ is a normal P( $\omega$ )-space if and only if the product space $X \times Y$ is normal for every separable metric space $Y$ . Thus suggests that any P(2)-space is a P( $\omega$ )-space. It seems to say that P(2) is identical to P( $\kappa$ ) where $\kappa$ is the countably infinite cardinal. The following theorem captures the idea.

Theorem 4
Let $\kappa$ be the positive integers $2,3,4,\cdots$ or $\kappa=\omega$ , the countably infinite cardinal. Let $X$ be a space. Then $X$ is a P(2)-space if and only if $X$ is a P( $\kappa$ )-space.

To give a context for Theorem 4, note that if $X$ is a P( $\kappa$ )-space, then $X$ is a P( $\tau$ )-space for any cardinal $\tau$ less than $\kappa$ . Thus if $X$ is a P(3)-space, then it is a P(2)-space and also a P(1)-space. In the definition of P( $\kappa$ )-space, the index set $\Gamma$ is the set of all finite sequences of elements of $\kappa$ . If the definition for P( $\kappa$ )-space holds, it would also hold for the index set consisting of finite sequences of elements of $\tau$ where $\tau<\kappa$ . Thus if the definition for P( $\omega$ )-space holds, it would hold for P( $n$ )-space for all integers $n$ .

Theorem 4 says that when the definition of P(2)-space holds, the definition would hold for all larger cardinals up to $\omega$ .

In light of Theorem 1 and Dowker's theorem, we have the following corollary. If the product of a space $X$ with every metric space is normal, then the product of $X$ with every compact metric space is normal.

Corollary 5
Let $X$ be a space. If $X$ is a normal P-space, then $X$ is a normal and countably paracompact space.

Examples of Normal P-Space

Here’s several classes of spaces that are normal P-spaces.

Metric spaces.
Compact spaces (link).
$\sigma$ -compact spaces (link).
Paracompact locally compact spaces (link).
Paracompact $\sigma$ -locally compact spaces (link).
Normal countably compact spaces (link).
Perfectly normal spaces (link).
$\Sigma$ -product of real lines.

Clearly any metric space is a normal P-space since the product of any two metric spaces is a metric space. Any compact space is a normal P-space since the product of a compact space and a paracompact space is paracompact, hence normal. For each of the classes of spaces listed above, the product with any metric space is normal. See the corresponding links for proofs of the key theorems.

The $\Sigma$ -product of real lines $\Sigma_{\alpha<\tau} \mathbb{R}$ is a normal P-space. For any metric space $Y$ , the product $(\Sigma_{\alpha<\tau} \mathbb{R}) \times Y$ is a $\Sigma$ -product of metric spaces. By a well known result, the $\Sigma$ -product of metric spaces is normal.

Examples of Non-Normal P-Spaces

Paracompact $\sigma$ -locally compact spaces are normal P-spaces since the product of such a space with any paracompact space is paracompact. However, the product of paracompact spaces in general is not normal. The product of Michael line (a hereditarily paracompact space) and the space of irrational numbers (a metric space) is not normal (discussed here). Thus the Michael line is not a normal P-space. More specifically the Michael line fails to be a normal P(2)-space. However, it is a normal P(1)-space (i.e. normal and countably paracompact space).

The Michael line is obtained from the usual real line topology by making the irrational points isolated. Instead of using the irrational numbers, we can obtain a similar space by making points in a Bernstein set isolated. The resulting space $X$ is a Michael line-like space. The product of $X$ with the starting Bernstein set (a subset of the real line with the usual topology) is not normal. Thus this is another example of a normal space that is not a P(2)-space. See here for the details of how this space is constructed.

To look for more examples, look for non-normal product $X \times Y$ where one factor is normal and the other is a metric space.

More Examples

Based on the characterization theorem of Morita, normal P-spaces are very productively normal. Normal P-spaces are well behaved when taking product with metrizable spaces. However, they are not well behaved when taking product with non-metrizable spaces. Let’s look at several examples.

Consider the Sorgenfrey line. It is perfectly normal. Thus the product of the Sorgenfrey line with any metric space is also perfectly normal, hence normal. It is well known that the square of the Sorgenfrey line is not normal.

The space $\omega_1$ of all countable ordinals is a normal and countably compact space, hence a normal P-space. However, the product of $\omega_1$ and some compact spaces are not normal. For example, $\omega_1 \times (\omega_1 +1)$ is not normal. Another example: $\omega_1 \times I^I$ is not normal where $I=[0,1]$ . The idea here is that the product of $\omega_1$ and any compact space with uncountable tightness is not normal (see here).

Compact spaces are normal P-spaces. As discussed in the preceding paragraph, the product of any compact space with uncountable tightness and the space $\omega_1$ is not normal.

Even as nice a space as the unit interval $[0,1]$ , it is not always productive. The product of $[0,1]$ with a Dowker space is not normal (see here).

In general, normality is not preserved in the product space operation. the best we can ask for is that normal spaces be productively normal with respect to a narrow class of spaces. For normal P-spaces, that narrow class of spaces is the class of metric spaces. However, normal product is not a guarantee outside of the productive class in question.

Reference

Morita K., On the Product of a Normal Space with a Metric Space, Proc. Japan Acad., Vol. 39, 148-150, 1963. (article information; paper)
Morita K., Products of Normal Spaces with Metric Spaces, Math. Ann., Vol. 154, 365-382, 1964.
Morita K., Nagata J., Topics in General Topology, Elsevier Science Publishers, B. V., The Netherlands, 1989.
Telgárski R., A characterization of P-spaces, Proc. Japan Acad., Vol. 51, 802–807, 1975.

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Dan Ma math

Daniel Ma mathematics

$\copyright$ 2018 – Dan Ma

In between G-delta diagonal and submetrizable

Posted on March 21, 2018 by Dan Ma

This post discusses the property of having a $G_\delta$ -diagonal and related diagonal properties. The focus is on the diagonal properties in between $G_\delta$ -diagonal and submetrizability. The discussion is followed by a diagram displaying the relative strengths of these properties. Some examples and questions are discussed.

G-delta Diagonal

In any space $Y$ , a subset $A$ is said to be a $G_\delta$ -set in the space $Y$ (or $A$ is a $G_\delta$ -subset of $Y$ ) if $A$ is the intersection of countably many open subsets of $Y$ . A subset $A$ of $Y$ is an $F_\sigma$ -set in $Y$ (or $A$ is an $F_\sigma$ -subset of $Y$ ) if $A$ is the union of countably closed subsets of the space $Y$ . Of course, the set $A$ is a $G_\delta$ -set if and only if $Y-A$ , the complement of $A$ , is an $F_\sigma$ -set.

The diagonal of the space $X$ is the set $\Delta=\{ (x,x): x \in X \}$ , which is a subset of the square $X \times X$ . When the set $\Delta$ is a $G_\delta$ -set in the space $X \times X$ , we say that the space $X$ has a $G_\delta$ -diagonal.

It is straightforward to verify that the space $X$ is a Hausdorff space if and only if the diagonal $\Delta$ is a closed subset of $X \times X$ . As a result, if $X$ is a Hausdorff space such that $X \times X$ is perfectly normal, then the diagonal would be a closed set and thus a $G_\delta$ -set. Such spaces, including metric spaces, would have a $G_\delta$ -diagonal. Thus any metric space has a $G_\delta$ -diagonal.

A space $X$ is submetrizable if there is a metrizable topology that is weaker than the topology for $X$ . Then the diagonal $\Delta$ would be a $G_\delta$ -set with respect to the weaker metrizable topology of $X \times X$ and thus with respect to the orginal topology of $X$ . This means that the class of spaces having $G_\delta$ -diagonals also include the submetrizable spaces. As a result, Sorgenfrey line and Michael line have $G_\delta$ -diagonals since the Euclidean topology are weaker than both topologies.

A space having a $G_\delta$ -diagonal is a simple topological property. Such spaces form a wide class of spaces containing many familiar spaces. According to the authors in [2], the property of having a $G_\delta$ -diagonal is an important ingredient of submetrizability and metrizability. For example, any compact space with a $G_\delta$ -diagonal is metrizable (see this blog post). Any paracompact or Lindelof space with a $G_\delta$ -diagonal is submetrizable. Spaces with $G_\delta$ -diagonals are also interesting in their own right. It is a property that had been research extensively. It is also a current research topic; see [7].

A Closer Look

To make the discussion more interesting, let’s point out a few essential definitions and notations. Let $X$ be a space. Let $\mathcal{U}$ be a collection of subsets of $X$ . Let $A \subset X$ . The notation $St(A, \mathcal{U})$ refers to the set $St(A, \mathcal{U})=\cup \{U \in \mathcal{U}: A \cap U \ne \varnothing \}$ . In other words, $St(A, \mathcal{U})$ is the union of all the sets in $\mathcal{U}$ that intersect the set $A$ . The set $St(A, \mathcal{U})$ is also called the star of the set $A$ with respect to the collection $\mathcal{U}$ .

If $A=\{ x \}$ , we write $St(x, \mathcal{U})$ instead of $St(\{ x \}, \mathcal{U})$ . Then $St(x, \mathcal{U})$ refers to the union of all sets in $\mathcal{U}$ that contain the point $x$ . The set $St(x, \mathcal{U})$ is then called the star of the point $x$ with respect to the collection $\mathcal{U}$ .

Note that the statement of $X$ having a $G_\delta$ -diagonal is defined by a statement about the product $X \times X$ . It is desirable to have a translation that is a statement about the space $X$ .

Theorem 1
Let $X$ be a space. Then the following statements are equivalent.

The space $X$ has a $G_\delta$ -diagonal.
There exists a sequence $\mathcal{U}_0,\mathcal{U}_1,\mathcal{U}_2,\cdots$ of open covers of $X$ such that for each $x \in X$ , $\{ x \}=\bigcap \{ St(x, \mathcal{U}_n): n=0,1,2,\cdots \}$ .

The sequence of open covers in condition 2 is called a $G_\delta$ -diagonal sequence for the space $X$ . According to condition 2, at any given point, the stars of the point with respect to the open covers in the sequence collapse to the given point.

One advantage of a $G_\delta$ -diagonal sequence is that it is entirely about points of the space $X$ . Thus we can work with such sequences of open covers of $X$ instead of the $G_\delta$ -set $\Delta$ in $X \times X$ . Theorem 1 is not a word for word translation. However, the proof is quote natural.

Suppose that $\Delta=\cap \{U_n: n=0,1,2,\cdots \}$ where each $U_n$ is an open subset of $X \times X$ . Then let $\mathcal{U}_n=\{U \subset X: U \text{ open and } U \times U \subset U_n \}$ . It can be verify that $\mathcal{U}_0,\mathcal{U}_1,\mathcal{U}_2,\cdots$ is a $G_\delta$ -diagonal sequence for $X$ .

Suppose that $\mathcal{U}_0,\mathcal{U}_1,\mathcal{U}_2,\cdots$ is a $G_\delta$ -diagonal sequence for $X$ . For each $n$ , let $U_n=\cup \{ U \times U: U \in \mathcal{U}_n \}$ . It follows that $\Delta=\bigcap_{n=0}^\infty U_n$ . $\square$

It is informative to compare the property of $G_\delta$ -diagonal with the definition of Moore spaces. A development for the space $X$ is a sequence $\mathcal{D}_0,\mathcal{D}_1,\mathcal{D}_2,\cdots$ of open covers of $X$ such that for each $x \in X$ , $\{ St(x, \mathcal{D}_n): n=0,1,2,\cdots \}$ is a local base at the point $x$ . A space is said to be developable if it has a development. The space $X$ is said to be a Moore space if $X$ is a Hausdorff and regular space that has a development.

The stars of a given point with respect to the open covers of a development form a local base at the given point, and thus collapse to the given point. Thus a development is also a $G_\delta$ -diagonal sequence. It then follows that any Moore space has a $G_\delta$ -diagonal.

A point in a space is a $G_\delta$ -point if the point is the intersection of countably many open sets. Then having a $G_\delta$ -diagonal sequence implies that that every point of the space is a $G_\delta$ -point since every point is the intersection of the stars of that point with respect to a $G_\delta$ -diagonal sequence. In contrast, any Moore space is necessarily a first countable space since the stars of any given point with respect to the development is a countable local base at the given point. The parallel suggests that spaces with $G_\delta$ -diagonals can be thought of as a weak form of Moore spaces (at least a weak form of developable spaces).

Regular G-delta Diagonal

We discuss other diagonal properties. The space $X$ is said to have a regular $G_\delta$ -diagonal if $\Delta=\cap \{\overline{U_n}:n=0,1,2,\cdots \}$ where each $U_n$ is an open subset of $X \times X$ such that $\Delta \subset U_n$ . This diagonal property also has an equivalent condition in terms of a diagonal sequence.

Theorem 2
Let $X$ be a space. Then the following statements are equivalent.

The space $X$ has a regular $G_\delta$ -diagonal.
There exists a sequence $\mathcal{U}_0,\mathcal{U}_1,\mathcal{U}_2,\cdots$ of open covers of $X$ such that for every two distinct points $x,y \in X$ , there exist open sets $U$ and $V$ with $x \in U$ and $y \in V$ and there also exists an $n$ such that no member of $\mathcal{U}_n$ intersects both $U$ and $V$ .

For convenience, we call the sequence described in Theorem 2 a regular $G_\delta$ -diagonal sequence. It is clear that if the diagonal of a space is a regular $G_\delta$ -diagonal, then it is a $G_\delta$ -diagonal. It can also be verified that a regular $G_\delta$ -diagonal sequence is also a $G_\delta$ -diagonal sequence. To see this, let $\mathcal{U}_0,\mathcal{U}_1,\mathcal{U}_2,\cdots$ be a regular $G_\delta$ -diagonal sequence for $X$ . Suppose that $y \ne x$ and $y \in \bigcap_k St(x, \mathcal{U}_k)$ . Choose open sets $U$ and $V$ and an integer $n$ guaranteed by the regular $G_\delta$ -diagonal sequence. Since $y \in St(x, \mathcal{U}_n)$ , choose $B \in \mathcal{U}_n$ such that $x,y \in B$ . Then $B$ would be an element of $\mathcal{U}_n$ that meets both $U$ and $V$ , a contradiction. Then $\{ x \}= \bigcap_k St(x, \mathcal{U}_k)$ for all $x \in X$ .

To proof Theorem 2, suppose that $X$ has a regular $G_\delta$ -diagonal. Let $\Delta=\bigcap_{k=0}^\infty \overline{U_k}$ where each $U_k$ is open in $X \times X$ and $\Delta \subset U_k$ . For each $k$ , let $\mathcal{U}_k$ be the collection of all open subsets $U$ of $X$ such that $U \times U \subset U_k$ . It can be verified that $\{ \mathcal{U}_k \}$ is a regular $G_\delta$ -diagonal sequence for $X$ .

On the other hand, suppose that $\{ \mathcal{U}_k \}$ is a regular $G_\delta$ -diagonal sequence for $X$ . For each $k$ , let $U_k=\cup \{U \times U: U \in \mathcal{U}_k \}$ . It can be verified that $\Delta=\bigcap_{k=0}^\infty \overline{U_k}$ . $\square$

Rank-k Diagonals

Metric spaces and submetrizable spaces have regular $G_\delta$ -diagonals. We discuss this fact after introducing another set of diagonal properties. First some notations. For any family $\mathcal{U}$ of subsets of the space $X$ and for any $x \in X$ , define $St^1(x, \mathcal{U})=St(x, \mathcal{U})$ . For any integer $k \ge 2$ , let $St^k(x, \mathcal{U})=St^{k-1}(St(x, \mathcal{U}))$ . Thus $St^{2}(x, \mathcal{U})$ is the star of the star $St(x, \mathcal{U})$ with respect to $\mathcal{U}$ and $St^{3}(x, \mathcal{U})$ is the star of $St^{2}(x, \mathcal{U})$ and so on.

Let $X$ be a space. A sequence $\mathcal{U}_0,\mathcal{U}_1,\mathcal{U}_2,\cdots$ of open covers of $X$ is said to be a rank- $k$ diagonal sequence of $X$ if for each $x \in X$ , we have $\{ x \}=\bigcap_{j=0}^\infty St^k(x,\mathcal{U}_j)$ . When the space $X$ has a rank- $k$ diagonal sequence, the space is said to have a rank- $k$ diagonal. Clearly a rank-1 diagonal sequence is simply a $G_\delta$ -diagonal sequence as defined in Theorem 1. Thus having a rank-1 diagonal is the same as having a $G_\delta$ -diagonal.

It is also clear that having a higher rank diagonal implies having a lower rank diagonal. This follows from the fact that a rank $k+1$ diagonal sequence is also a rank $k$ diagonal sequence.

The following lemma builds intuition of the rank- $k$ diagonal sequence. For any two distinct points $x$ and $y$ of a space $X$ , and for any integer $d \ge 2$ , a $d$ -link path from $x$ to $y$ is a set of open sets $W_1,W_2,\cdots,W_d$ such that $x \in W_1$ , $y \in W_d$ and $W_t \cap W_{t+1} \ne \varnothing$ for all $t=1,2,\cdots,d-1$ . By default, a single open set $W$ containing both $x$ and $y$ is a d-link path from $x$ to $y$ for any integer $d \ge 1$ .

Lemma 3
Let $X$ be a space. Let $k$ be a positive integer. Let $\mathcal{U}_0,\mathcal{U}_1,\mathcal{U}_2,\cdots$ be a sequence of open covers of $X$ . Then the following statements are equivalent.

The sequence $\mathcal{U}_0,\mathcal{U}_1,\mathcal{U}_2,\cdots$ is a rank- $k$ diagonal sequence for the space $X$ .
For any two distinct points $x$ and $y$ of $X$ , there is an integer $n$ such that $y \notin St^k(x,\mathcal{U}_n)$ .
For any two distinct points $x$ and $y$ of $X$ , there is an integer $n$ such that there is no $k$ -link path from $x$ to $y$ consisting of elements of $\mathcal{U}_n$ .

It can be seen directly from definition that Condition 1 and Condition 2 are equivalent. For Condition 3, observe that the set $St^k(x,\mathcal{U}_n)$ is the union of $k$ types of open sets – open sets in $\mathcal{U}_n$ containing $x$ , open sets in $\mathcal{U}_n$ that intersect the first type, open sets in $\mathcal{U}_n$ that intersect the second type and so on down to the open sets in $\mathcal{U}_n$ that intersect $St^{k-1}(x,\mathcal{U}_n)$ . A path is formed by taking one open set from each type.

We now show a few basic results that provide further insight on the rank- $k$ diagonal.

Theorem 4
Let $X$ be a space. If the space $X$ has a rank-2 diagonal, then $X$ is a Hausdorff space.

Theorem 5
Let $X$ be a Moore space. Then $X$ has a rank-2 diagonal.

Theorem 6
Let $X$ be a space. If $X$ has a rank-3 diagonal, then $X$ has a regular $G_\delta$ -diagonal.

Once Lemma 3 is understood, Theorem 4 is also easily understood. If a space $X$ has a rank-2 diagonal sequence $\{ \mathcal{U}_n \}$ , then for any two distinct points $x$ and $y$ , we can always find an $n$ where there is no 2-link path from $x$ to $y$ . Then $x$ and $y$ can be separated by open sets in $\mathcal{U}_n$ . Thus these diagonal ranking properties confer separation axioms. We usually start off a topology discussion by assuming a reasonable separation axiom (usually implicitly). The fact that the diagonal ranking gives a bonus makes it even more interesting. Apparently many authors agree since $G_\delta$ -diagonal and related topics had been researched extensively over decades.

To prove Theorem 5, let $\{ \mathcal{U}_n \}$ be a development for the space $X$ . Let $x$ and $y$ be two distinct points of $X$ . We claim that there exists some $n$ such that $y \notin St^2(x,\mathcal{U}_n)$ . Suppose not. This means that for each $n$ , $y \in St^2(x,\mathcal{U}_n)$ . This also means that $St(x,\mathcal{U}_n) \cap St(y,\mathcal{U}_n) \ne \varnothing$ for each $n$ . Choose $x_n \in St(x,\mathcal{U}_n) \cap St(y,\mathcal{U}_n)$ for each $n$ . Since $X$ is a Moore space, $\{ St(x,\mathcal{U}_n) \}$ is a local base at $x$ . Then $\{ x_n \}$ converges to $x$ . Since $\{ St(y,\mathcal{U}_n) \}$ is a local base at $y$ , $\{ x_n \}$ converges to $y$ , a contradiction. Thus the claim that there exists some $n$ such that $y \notin St^2(x,\mathcal{U}_n)$ is true. By Lemma 3, a development for a Moore space is a rank-2 diagonal sequence.

To prove Theorem 6, let $\{ \mathcal{U}_n \}$ be a rank-3 diagonal sequence for the space $X$ . We show that $\{ \mathcal{U}_n \}$ is also a regular $G_\delta$ -diagonal sequence for $X$ . Suppose $x$ and $y$ are two distinct points of $X$ . By Lemma 3, there exists an $n$ such that there is no 3-link path consisting of open sets in $\mathcal{U}_n$ that goes from $x$ to $y$ . Choose $U \in \mathcal{U}_n$ with $x \in U$ . Choose $V \in \mathcal{U}_n$ with $y \in V$ . Then it follows that no member of $\mathcal{U}_n$ can intersect both $U$ and $V$ (otherwise there would be a 3-link path from $x$ to $y$ ). Thus $\{ \mathcal{U}_n \}$ is also a regular $G_\delta$ -diagonal sequence for $X$ .

We now show that metric spaces have rank- $k$ diagonal for all integer $k \ge 1$ .

Theorem 7
Let $X$ be a metrizable space. Then $X$ has rank- $k$ diagonal for all integers $k \ge 1$ .

If $d$ is a metric that generates the topology of $X$ , and if $\mathcal{U}_n$ is the collection of all open subsets with diameters $\le 2^{-n}$ with respect to the metrix $d$ then $\{ \mathcal{U}_n \}$ is a rank- $k$ diagonal sequence for $X$ for any integer $k \ge 1$ .

We instead prove Theorem 7 topologically. To this end, we use an appropriate metrization theorem. The following theorem is a good candidate.

Alexandrov-Urysohn Metrization Theorem. A space $X$ is metrizable if and only if the space $X$ has a development $\{ \mathcal{U}_n \}$ such that for any $U_1,U_2 \in \mathcal{U}_{n+1}$ with $U_1 \cap U_2 \ne \varnothing$ , the set $U_1 \cup U_2$ is contained in some element of $\mathcal{U}_n$ . See Theorem 1.5 in p. 427 of [5].

Let $\{ \mathcal{U}_n \}$ be the development from Alexandrov-Urysohn Metrization Theorem. It is a development with a strong property. Each open cover in the development refines the preceding open cover in a special way. This refinement property allows us to show that it is a rank- $k$ diagonal sequence for $X$ for any integer $k \ge 1$ .

First, we make a few observations about $\{ \mathcal{U}_n \}$ . From the statement of the theorem, each $\mathcal{U}_{n+1}$ is a refinement of $\mathcal{U}_n$ . As a result of this observation, $\mathcal{U}_{m}$ is a refinement of $\mathcal{U}_n$ for any $m>n$ . Furthermore, for each $x \in X$ , $\text{St}(x,\mathcal{U}_m) \subset \text{St}(x,\mathcal{U}_n)$ for any $m>n$ .

Let $x, y \in X$ with $x \ne y$ . Based on the preceding observations, it follows that there exists some $m$ such that $\text{St}(x,\mathcal{U}_m) \cap \text{St}(y,\mathcal{U}_m)=\varnothing$ . We claim that there exists some integer $h>m$ such that there are no $k$ -link path from $x$ to $y$ consisting of open sets from $\mathcal{U}_h$ . Then $\{ \mathcal{U}_n \}$ is a rank- $k$ diagonal sequence for $X$ according to Lemma 3.

We show this claim is true for $k=2$ . Observe that there cannot exist $U_1, U_2 \in \mathcal{U}_{m+1}$ such that $x \in U_1$ , $y \in U_2$ and $U_1 \cap U_2 \ne \varnothing$ . If there exists such a pair, then $U_1 \cup U_2$ would be contained in $\text{St}(x,\mathcal{U}_m)$ and $\text{St}(y,\mathcal{U}_m)$ , a contradiction. Putting it in another way, there cannot be any 2-link path $U_1,U_2$ from $x$ to $y$ such that the open sets in the path are from $\mathcal{U}_{m+1}$ . According to Lemma 3, the sequence $\{ \mathcal{U}_n \}$ is a rank-2 diagonal sequence for the space $X$ .

In general for any $k \ge 2$ , there cannot exist any $k$ -link path $U_1,\cdots,U_k$ from $x$ to $y$ such that the open sets in the path are from $\mathcal{U}_{m+k-1}$ . The argument goes just like the one for the case for $k=2$ . Suppose the path $U_1,\cdots,U_k$ exists. Using the special property of $\{ \mathcal{U}_n \}$ , the 2-link path $U_1,U_2$ is contained in some open set in $\mathcal{U}_{m+k-2}$ . The path $U_1,\cdots,U_k$ is now contained in a $(k-1)$ -link path consisting of elements from the open cover $\mathcal{U}_{m+k-2}$ . Continuing the refinement process, the path $U_1,\cdots,U_k$ is contained in a 2-link path from $x$ to $y$ consisting of elements from $\mathcal{U}_{m+1}$ . Like before this would lead to a contradiction. According to Lemma 3, $\{ \mathcal{U}_n \}$ is a rank- $k$ diagonal sequence for the space $X$ for any integer $k \ge 2$ .

Of course, any metric space already has a $G_\delta$ -diagonal. We conclude that any metrizable space has a rank- $k$ diagonal for any integer $k \ge 1$ . $\square$

We have the following corollary.

Corollary 8
Let $X$ be a submetrizable space. Then $X$ has rank- $k$ diagonal for all integer $k \ge 1$ .

In a submetrizable space, the weaker metrizable topology has a rank- $k$ diagonal sequence, which in turn is a rank- $k$ diagonal sequence in the original topology.

Examples and Questions

The preceding discussion focuses on properties that are in between $G_\delta$ -diagonal and submetrizability. In fact, one of the properties has infinitely many levels (rank- $k$ diagonal for integers $k \ge 1$ ). We would like to have a diagram showing the relative strengths of these properties. Before we do so, consider one more diagonal property.

Let $X$ be a space. The set $A \subset X$ is said to be a zero-set in $X$ if there is a continuous $f:X \rightarrow [0,1]$ such that $A=f^{-1}(0)$ . In other words, a zero-set is a set that is the inverse image of zero for some continuous real-valued function defined on the space in question.

A space $X$ has a zero-set diagonal if the diagonal $\Delta=\{ (x,x): x \in X \}$ is a zero-set in $X \times X$ . The space $X$ having a zero-set diagonal implies that $X$ has a regular $G_\delta$ -diagonal, and thus a $G_\delta$ -diagonal. To see this, suppose that $\Delta=f^{-1}(0)$ where $f:X \times X \rightarrow [0,1]$ is continuous. Then $\Delta=\bigcap_{n=1}^\infty \overline{U_n}$ where $U_n=f^{-1}([0,1/n))$ . Thus having a zero-set diagonal is a strong property.

We have the following diagram.

The diagram summarizes the preceding discussion. From top to bottom, the stronger properties are at the top. From left to right, the stronger properties are on the left. The diagram shows several properties in between $G_\delta$ -diagonal at the bottom and submetrizability at the top.

Note that the statement at the very bottom is not explicitly a diagonal property. It is placed at the bottom because of the classic result that any compact space with a $G_\delta$ -diagonal is metrizable.

In the diagram, “rank-k diagonal” means that the space has a rank- $k$ diagonal where $k \ge 1$ is an integer, which in terms means that the space has a rank- $k$ diagonal sequence as defined above. Thus rank- $k$ diagonal is not to be confused with the rank of a diagonal. The rank of the diagonal of a given space is the largest integer $k$ such that the space has a rank- $k$ diagonal. For example, for a space that has a rank-2 diagonal but has no rank-3 diagonal, the rank of the diagonal is 2.

To further make sense of the diagram, let’s examine examples.

The Mrowka space is a classic example of a space with a $G_\delta$ -diagonal that is not submetrizable (introduced here). Where is this space located in the diagram? The Mrowka space, also called Psi-space, is defined using a maximal almost disjoint family of subsets of $\omega$ . We denote such a space by $\Psi(\mathcal{A})$ where $\mathcal{A}$ is a maximal almost disjoint family of subsets of $\omega$ . It is a pseudocompact Moore space that is not submetrizable. As a Moore space, it has a rank-2 diagonal sequence. A well known result states that any pseudocompact space with a regular $G_\delta$ -diagonal is metrizable (see here). As a non-submetrizable space, the Mrowka space cannot have a regular $G_\delta$ -diagonal. Thus $\Psi(\mathcal{A})$ is an example of a space with a rank-2 diagonal but not a rank-3 diagonal sequence.

Examples of non-submetrizable spaces with stronger diagonal properties are harder to come by. We discuss examples that are found in the literature.

Example 2.9 in [2] is a Tychonoff separable Moore space $Z$ that has a rank-3 diagonal but not of higher diagonal rank. As a result of not having a rank-4 diagonal, $Z$ is not submetrizable. Thus $Z$ is an example of a space with rank-3 diagonal (hence with a regular $G_\delta$ -diagonal) that is not submetrizable. According to a result in [6], any separable space with a zero-set diagonal is submetrizable. Then the space $Z$ is an example of a space with a regular $G_\delta$ -diagonal that does not have a zero-set diagonal. In fact, the authors of [2] indicated that this is the first such example.

Example 2.9 of [2] shows that having a rank-3 diagonal does not imply having a zero-set diagonal. If a space is strengthened to have a rank-4 diagonal, does it imply having a zero-set diagonal? This is essentially Problem 2.13 in [2].

On the other hand, having a rank-3 diagonal implies a rank-2 diagonal. If we weaken the hypothesis to just having a regular regular $G_\delta$ -diagonal, does it imply having a rank-2 diagonal? This is essentially Problem 2.14 in [2].

The authors of [2] conjectured that for each $n$ , there exists a space $X_n$ with a rank- $n$ diagonal but not having a rank- $(n+1)$ diagonal. This conjecture was answered affirmatively in [8] by constructing, for each integer $k \ge 4$ , a Tychonoff space with a rank- $k$ diagonal but not having a rank- $(k+1)$ diagonal. Thus even for high $k$ , a non-submetrizable space can be found with rank- $k$ diagonal.

One natural question is this. Is there a non-submetrizable space that has rank- $k$ diagonal for all $k \ge 1$ ? We have not seen this question stated in the literature. But it is clearly a natural question.

Example 2.17 in [2] is a non-submetrizable Moore space that has a zero-set diagonal and has rank-3 diagonal exactly (i.e. it does not have a higher rank diagonal). This example shows that having a zero-set diagonal does not imply having a rank-4 diagonal. A natural question is then this. Does having a zero-set diagonal imply having a rank-3 diagonal? This appears to be an open question. This is hinted by Problem 2.19 in [2]. It asks, if $X$ is a normal space with a zero-set diagonal, does $X$ have at least a rank-2 diagonal?

The property of having a $G_\delta$ -diagonal and related properties is a topic that had been researched extensively over the decades. It is still an active topic of research. The discussion in this post only touches on the surface. There are many other diagonal properties not covered here. To further investigate, check with the papers listed below and also consult with information available in the literature.

Reference

Arhangelskii A. V., Burke D. K., Spaces with a regular $G_\delta$ -diagonal, Topology and its Applications, Vol. 153, No. 11, 1917–1929, 2006.
Arhangelskii A. V., Buzyakova R. Z., The rank of the diagonal and submetrizability, Comment. Math. Univ. Carolinae, Vol. 47, No. 4, 585-597, 2006.
Buzyakova R. Z., Cardinalities of ccc-spaces with regular $G_\delta$ -diagonals, Topology and its Applications, Vol. 153, 1696–1698, 2006.
Buzyakova R. Z., Observations on spaces with zeroset or regular $G_\delta$ -diagonals, Comment. Math. Univ. Carolinae, Vol. 46, No. 3, 469-473, 2005.
Gruenhage, G., Generalized Metric Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 423-501, 1984.
Martin H. W., Contractibility of topological spaces onto metric spaces, Pacific J. Math., Vol. 61, No. 1, 209-217, 1975.
Xuan Wei-Feng, Shi Wei-Xue, On spaces with rank k-diagonals or zeroset diagonals, Topology Proceddings, Vol. 51, 245{251, 2018.
Yu Zuoming, Yun Ziqiu, A note on the rank of diagonals, Topology and its Applications, Vol. 157, 1011–1014, 2010.

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Dan Ma math

Daniel Ma mathematics

$\copyright$ 2018 – Dan Ma

Pseudocompact spaces with regular G-delta diagonals

Posted on February 19, 2018 by Dan Ma

This post complements two results discussed in two previous blog posts concerning $G_\delta$ -diagonal. One result is that any compact space with a $G_\delta$ -diagonal is metrizable (see here). The other result is that the compactness in the first result can be relaxed to countably compactness. Thus any countably compact space with a $G_\delta$ -diagonal is metrizable (see here). The countably compactness in the second result cannot be relaxed to pseudocompactness. The Mrowka space is a pseudocompact space with a $G_\delta$ -diagonal that is not submetrizable, hence not metrizable (see here). However, if we strengthen the $G_\delta$ -diagonal to a regular $G_\delta$ -diagonal while keeping pseudocompactness fixed, then we have a theorem. We prove the following theorem.

Theorem 1
If the space $X$ is pseudocompact and has a regular $G_\delta$ -diagonal, then $X$ is metrizable.

All spaces are assumed to be Hausdorff and completely regular. The assumption of completely regular is crucial. The proof of Theorem 1 relies on two lemmas concerning pseudocompact spaces (one proved in a previous post and one proved here). These two lemmas work only for completely regular spaces.

The proof of Theorem 1 uses a metrization theorem. The best metrization to use in this case is Moore metrization theorem (stated below). The result in Theorem 1 is found in [2].

First some basics. Let $X$ be a space. The diagonal of the space $X$ is the set $\Delta=\{ (x,x): x \in X \}$ . When the diagonal $\Delta$ , as a subset of $X \times X$ , is a $G_\delta$ -set, i.e. $\Delta$ is the intersection of countably many open subsets of $X \times X$ , the space $X$ is said to have a $G_\delta$ -diagonal.

The space $X$ is said to have a regular $G_\delta$ -diagonal if the diagonal $\Delta$ is a regular $G_\delta$ -set in $X \times X$ , i.e. $\Delta=\bigcap_{n=1}^\infty \overline{U_n}$ where each $U_n$ is an open subset of $X \times X$ with $\Delta \subset U_n$ . If $\Delta=\bigcap_{n=1}^\infty \overline{U_n}$ , then $\Delta=\bigcap_{n=1}^\infty \overline{U_n}=\bigcap_{n=1}^\infty U_n$ . Thus if a space has a regular $G_\delta$ -diagonal, it has a $G_\delta$ -diagonal. We will see that there exists a space with a $G_\delta$ -diagonal that fails to be a regular $G_\delta$ -diagonal.

The space $X$ is a pseudocompact space if for every continuous function $f:X \rightarrow \mathbb{R}$ , the image $f(X)$ is a bounded set in the real line $\mathbb{R}$ . Pseudocompact spaces are discussed in considerable details in this previous post. We will rely on results from this previous post to prove Theorem 1.

The following lemma is used in proving Theorem 1.

Lemma 2
Let $X$ be a pseudocompact space. Suppose that $O_1,O_2,O_2,\cdots$ is a decreasing sequence of non-empty open subsets of $X$ such that $\bigcap_{n=1}^\infty O_n=\bigcap_{n=1}^\infty \overline{O_n}=\{ x \}$ for some point $x \in X$ . Then $\{ O_n \}$ is a local base at the point $x$ .

Proof of Lemma 2
Let $O_1,O_2,O_2,\cdots$ be a decreasing sequence of open subsets of $X$ such that $\bigcap_{n=1}^\infty O_n=\bigcap_{n=1}^\infty \overline{O_n}=\{ x \}$ . Let $U$ be open in $X$ with $x \in U$ . If $O_n \subset U$ for some $n$ , then we are done. Suppose that $O_n \not \subset U$ for each $n$ .

Choose open $V$ with $x \in V \subset \overline{V} \subset U$ . Consider the sequence $\{ O_n \cap (X-\overline{V}) \}$ . This is a decreasing sequence of non-empty open subsets of $X$ . By Theorem 2 in this previous post, $\bigcap \overline{O_n \cap (X-\overline{V})} \ne \varnothing$ . Let $y$ be a point in this non-empty set. Note that $y \in \bigcap_{n=1}^\infty \overline{O_n}$ . This means that $y=x$ . Since $x \in \overline{O_n \cap (X-\overline{V})}$ for each $n$ , any open set containing $x$ would contain a point not in $\overline{V}$ . This is a contradiction since $x \in V$ . Thus it must be the case that $x \in O_n \subset U$ for some $n$ . $\square$

The following metrization theorem is useful in proving Theorem 1.

Theorem 3 (Moore Metrization Theorem)
Let $X$ be a space. Then $X$ is metrizable if and only if the following condition holds.

There exists a decreasing sequence $\mathcal{B}_1,\mathcal{B}_2,\mathcal{B}_3,\cdots$ of open covers of $X$ such that for each $x \in X$ , the sequence $\{ St(St(x,\mathcal{B}_n),\mathcal{B}_n):n=1,2,3,\cdots \}$ is a local base at the point $x$ .

For any family $\mathcal{U}$ of subsets of $X$ , and for any $A \subset X$ , the notation $St(A,\mathcal{U})$ refers to the set $\cup \{U \in \mathcal{U}: U \cap A \ne \varnothing \}$ . In other words, it is the union of all sets in $\mathcal{U}$ that contain points of $A$ . The set $St(A,\mathcal{U})$ is also called the star of the set $A$ with respect to the family $\mathcal{U}$ . If $A=\{ x \}$ , we write $St(x,\mathcal{U})$ instead of $St(\{ x \},\mathcal{U})$ . The set $St(St(x,\mathcal{B}_n),\mathcal{B}_n)$ indicated in Theorem 3 is the star of the set $St(x,\mathcal{B}_n)$ with respect to the open cover $\mathcal{B}_n$ .

Theorem 3 follows from Theorem 1.4 in [1], which states that for any $T_0$ -space $X$ , $X$ is metrizable if and only if there exists a sequence $\mathcal{G}_1, \mathcal{G}_2, \mathcal{G}_3,\cdots$ of open covers of $X$ such that for each open $U \subset X$ and for each $x \in U$ , there exist an open $V \subset X$ and an integer $n$ such that $x \in V$ and $St(V,\mathcal{G}_n) \subset U$ .

Proof of Theorem 1

Suppose $X$ is pseudocompact such that its diagonal $\Delta=\bigcap_{n=1}^\infty \overline{U_n}$ where each $U_n$ is an open subset of $X \times X$ with $\Delta \subset U_n$ . We can assume that $U_1 \supset U_2 \supset \cdots$ . For each $n \ge 1$ , define the following:

$\mathcal{U}_n=\{ U \subset X: U \text{ open in } X \text{ and } U \times U \subset U_n \}$

Note that each $\mathcal{U}_n$ is an open cover of $X$ . Also note that $\{ \mathcal{U}_n \}$ is a decreasing sequence since $\{ U_n \}$ is a decreasing sequence of open sets. We show that $\{ \mathcal{U}_n \}$ is a sequence of open covers of $X$ that satisfies Theorem 3. We establish this by proving the following claims.

Claim 1. For each $x \in X$ , $\bigcap_{n=1}^\infty \overline{St(x,\mathcal{U}_n)}=\{ x \}$ .

To prove the claim, let $x \ne y$ . There is an integer $n$ such that $(x,y) \notin \overline{U_n}$ . Choose open sets $U$ and $V$ such that $(x,y) \in U \times V$ and $(U \times V) \cap \overline{U_n}=\varnothing$ . Note that $(x,y) \notin U_k$ and $(U \times V) \cap U_n=\varnothing$ .

We want to show that $V \cap St(x,\mathcal{U}_n)=\varnothing$ , which implies that $y \notin \overline{St(x,\mathcal{U}_n)}$ . Suppose $V \cap St(x,\mathcal{U}_n) \ne \varnothing$ . This means that $V \cap W \ne \varnothing$ for some $W \in \mathcal{U}_n$ with $x \in W$ . Then $(U \times V) \cap (W \times W) \ne \varnothing$ . Note that $W \times W \subset U_n$ . This implies that $(U \times V) \cap U_n \ne \varnothing$ , a contradiction. Thus $V \cap St(x,\mathcal{U}_n)=\varnothing$ . Since $y \in V$ , $y \notin \overline{St(x,\mathcal{U}_n)}$ . We have established that for each $x \in X$ , $\bigcap_{n=1}^\infty \overline{St(x,\mathcal{U}_n)}=\{ x \}$ .

Claim 2. For each $x \in X$ , $\{ St(x,\mathcal{U}_n) \}$ is a local base at the point $x$ .

Note that $\{ St(x,\mathcal{U}_n) \}$ is a decreasing sequence of open sets such that $\bigcap_{n=1}^\infty \overline{St(x,\mathcal{U}_n)}=\{ x \}$ . By Lemma 2, $\{ St(x,\mathcal{U}_n) \}$ is a local base at the point $x$ .

Claim 3. For each $x \in X$ , $\bigcap_{n=1}^\infty \overline{St(St(x,\mathcal{U}_n),\mathcal{U}_n)}=\{ x \}$ .

Let $x \ne y$ . There is an integer $n$ such that $(x,y) \notin \overline{U_n}$ . Choose open sets $U$ and $V$ such that $(x,y) \in U \times V$ and $(U \times V) \cap \overline{U_n}=\varnothing$ . It follows that $(U \times V) \cap \overline{U_t}=\varnothing$ for all $t \ge n$ . Furthermore, $(U \times V) \cap U_t=\varnothing$ for all $t \ge n$ . By Claim 2, choose integers $i$ and $j$ such that $St(x,\mathcal{U}_i) \subset U$ and $St(y,\mathcal{U}_j) \subset V$ . Choose an integer $k \ge \text{max}(n,i,j)$ . It follows that $(St(x,\mathcal{U}_i) \times St(y,\mathcal{U}_j)) \cap U_k=\varnothing$ . Since $\mathcal{U}_k \subset \mathcal{U}_i$ and $\mathcal{U}_k \subset \mathcal{U}_j$ , it follows that $(St(x,\mathcal{U}_k) \times St(y,\mathcal{U}_k)) \cap U_k=\varnothing$ .

We claim that $St(y,\mathcal{U}_k) \cap St(St(x,\mathcal{U}_k), \mathcal{U}_k)=\varnothing$ . Suppose not. Choose $w \in St(y,\mathcal{U}_k) \cap St(St(x,\mathcal{U}_k), \mathcal{U}_k)$ . It follows that $w \in B$ for some $B \in \mathcal{U}_k$ such that $B \cap St(x,\mathcal{U}_k) \ne \varnothing$ and $B \cap St(y,\mathcal{U}_k) \ne \varnothing$ . Furthermore $(St(x,\mathcal{U}_k) \times St(y,\mathcal{U}_k)) \cap (B \times B)=\varnothing$ . Note that $B \times B \subset U_k$ . This means that $(St(x,\mathcal{U}_k) \times St(y,\mathcal{U}_k)) \cap U_k \ne \varnothing$ , contradicting the fact observed in the preceding paragraph. It must be the case that $St(y,\mathcal{U}_k) \cap St(St(x,\mathcal{U}_k), \mathcal{U}_k)=\varnothing$ .

Because there is an open set containing $y$ , namely $St(y,\mathcal{U}_k)$ , that contains no points of $St(St(x,\mathcal{U}_k), \mathcal{U}_k)$ , $y \notin \overline{St(St(x,\mathcal{U}_n),\mathcal{U}_n)}$ . Thus Claim 3 is established.

Claim 4. For each $x \in X$ , $\{ St(St(x,\mathcal{U}_n),\mathcal{U}_n)) \}$ is a local base at the point $x$ .

Note that $\{ St(St(x,\mathcal{U}_n),\mathcal{U}_n) \}$ is a decreasing sequence of open sets such that $\bigcap_{n=1}^\infty \overline{St(St(x,\mathcal{U}_n),\mathcal{U}_n))}=\{ x \}$ . By Lemma 2, $\{ St(St(x,\mathcal{U}_n),\mathcal{U}_n) \}$ is a local base at the point $x$ .

In conclusion, the sequence $\mathcal{U}_1,\mathcal{U}_2,\mathcal{U}_3,\cdots$ of open covers satisfies the properties in Theorem 3. Thus any pseudocompact space with a regular $G_\delta$ -diagonal is metrizable. $\square$

Example

Any submetrizable space has a $G_\delta$ -diagonal. The converse is not true. A classic example of a non-submetrizable space with a $G_\delta$ -diagonal is the Mrowka space (discussed here). The Mrowka space is also called the psi-space since it is sometimes denoted by $\Psi(\mathcal{A})$ where $\mathcal{A}$ is a maximal family of almost disjoint subsets of $\omega$ . Actually $\Psi(\mathcal{A})$ would be a family of spaces since $\mathcal{A}$ is any maximal almost disjoint family. For any maximal $\mathcal{A}$ , $\Psi(\mathcal{A})$ is a pseudocompact non-submetrizable space that has a $G_\delta$ -diagonal. This example shows that the requirement of a regular $G_\delta$ -diagonal in Theorem 1 cannot be weakened to a $G_\delta$ -diagonal. See here for a more detailed discussion of this example.

Reference

Gruenhage, G., Generalized Metric Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 423-501, 1984.
McArthur W. G., $G_\delta$ -Diagonals and Metrization Theorems, Pacific Journal of Mathematics, Vol. 44, No. 2, 613-317, 1973.

$\text{ }$

Dan Ma math

Daniel Ma mathematics

$\copyright$ 2018 – Dan Ma

Drawing more Sorgenfrey continuous functions

Posted on January 1, 2018 by Dan Ma

In this previous post, we draw continuous functions on the Sorgenfrey line $S$ to gain insight about the function $C_p(S)$ . In this post, we draw more continuous functions with the goal of connecting $C_p(S)$ and $C_p(D)$ where $D$ is the double arrow space. For example, $C_p(D)$ can be embedded as a subspace of $C_p(S)$ . More interestingly, both function spaces $C_p(D)$ and $C_p(S)$ share the same closed and discrete subspace of cardinality continuum. As a result, the function space $C_p(D)$ is not normal.

Double Arrow Space

The underlying set for the double arrow space is $D=[0,1] \times \{ 0,1 \}$ , which is a subset in the Euclidean plane.

Figure 1 – The Double Arrow Space

The name of double arrow comes from the fact that an open neighborhood of a point in the upper line segment points to the right while an open neighborhood of a point in the lower line segment points to the left. This is demonstrated in the following diagram.

Figure 2 – Open Neighborhoods in the Double Arrow Space

More specifically, for any $a$ with $0 \le a < 1$ , a basic open set containing the point $(a,1)$ is of the form $\displaystyle \biggl[ [a,b) \times \{ 1 \} \biggr] \cup \biggl[ (a,b) \times \{ 0 \} \biggr]$ , painted red in Figure 2. One the other hand, for any $a$ with $0<a \le 1$ , a basic open set containing the point $(a,0)$ is of the form $\biggl[ (c,a) \times \{ 1 \} \biggr] \cup \biggl[ (c,a] \times \{ 0 \} \biggr]$ , painted blue in Figure 2. The upper right point $(1,1)$ and the lower left point $(0,0)$ are made isolated points.

The double arrow space is a compact space that is perfectly normal and not metrizable. Basic properties of this space, along with those of the lexicographical ordered space, are discussed in this previous post.

The drawing of continuous functions in this post aims to show the following results.

The function space $C_p(D)$ can be embedded as a subspace in the function $C_p(S)$ .
Both function spaces $C_p(D)$ and $C_p(S)$ share the same closed and discrete subspace of cardinality continuum.
The function space $C_p(D)$ is not normal.

Drawing a Map from Sorgenfrey Line onto Double Arrow Space

In order to show that $C_p(D)$ can be embedded into $C_p(S)$ , we draw a continuous map from the Sorgenfrey line $S$ onto the double arrow space $D$ . The following diagram gives the essential idea of the mapping we need.

Figure 3 – Mapping Sorgenfrey Line onto Double Arrow Space

The mapping shown in Figure 3 is to map the interval $[0,1]$ onto the upper line segment of the double arrow space, as demonstrated by the red arrow. Thus $x \mapsto (x,1)$ for any $x$ with $0 \le x \le 1$ . Essentially on the interval $[0,1]$ , the mapping is the identity map.

On the other hand, the mapping is to map the interval $[-1,0)$ onto the lower line segment of the double arrow space less the point $(0, 0)$ , as demonstrated by the blue arrow in Figure 3. Thus $-x \mapsto (x,0)$ for any $-x$ with $0<x \le 1$ . Essentially on the interval $[-1,0)$ , the mapping is the identity map times -1.

The mapping described by Figure 3 only covers the interval $[-1,1]$ in the domain. To complete the mapping, let $x \mapsto (1,1)$ for any $x \in (1, \infty)$ and $x \mapsto (0,0)$ for any $x \in (-\infty, -1)$ .

Let $h$ be the mapping that has been described. It maps the Sorgenfrey line onto the double arrow space. It is straightforward to verify that the map $h: S \rightarrow D$ is continuous.

Embedding

We use the following fact to show that $C_p(D)$ can be embedded into $C_p(S)$ .

Suppose that the space $Y$ is a continuous image of the space $X$ . Then $C_p(Y)$ can be embedded into $C_p(X)$ .

Based on this result, $C_p(D)$ can be embedded into $C_p(S)$ . The embedding that makes this true is $E(f)=f \circ h$ for each $f \in C_p(D)$ . Thus each function $f$ in $C_p(D)$ is identified with the composition $f \circ h$ where $h$ is the map defined in Figure 3. The fact that $E(f)$ is an embedding is shown in this previous post (see Theorem 1).

Same Closed and Discrete Subspace in Both Function Spaces

The following diagram describes a closed and discrete subspace of $C_p(S)$ .

Figure 4 – a family of Sorgenfrey continuous functions

For each $0<a<1$ , let $f_a: S \rightarrow \{0,1 \}$ be the continuous function described in Figure 4. The previous post shows that the set $F=\{ f_a: 0<a<1 \}$ is a closed and discrete subspace of $C_p(S)$ . We claim that $F \subset C_p(D) \subset C_p(S)$ .

To see that $F \subset C_p(D)$ , we define continuous functions $U_a: D \rightarrow \{0,1 \}$ such that $f_a=U_a \circ h$ . We can actually back out the map $U_a$ from $f_a$ in Figure 4 and the mapping $h$ . Here’s how. The function $f_a$ is piecewise constant (0 or 1). Let’s focus on the interval $[-1,1]$ in the domain of $f_a$ .

Consider where the function $f_a$ maps to the value 1. There are two intervals, $[a,1)$ and $[-1,-a)$ , where $f_a$ maps to 1. The mapping $h$ maps $[a,1)$ to the set $[a,1) \times \{ 1 \}$ . So the function $U_a$ must map $[a,1) \times \{ 1 \}$ to the value 1. The mapping $h$ maps $[-1,-a)$ to the set $(a,1] \times \{ 0 \}$ . So $U_a$ must map $(a,1] \times \{ 0 \}$ to the value 1.

Now consider where the function $f_a$ maps to the value 0. There are two intervals, $[0,a)$ and $[-a,0)$ , where $f_a$ maps to 0. The mapping $h$ maps $[0,a)$ to the set $[0,a) \times \{ 1 \}$ . So the function $U_a$ must map $[0,a) \times \{ 1 \}$ to the value 0. The mapping $h$ maps $[-a,0)$ to the set $(0,a] \times \{ 0 \}$ . So $U_a$ must map $(0,a] \times \{ 0 \}$ to the value 0.

To take care of the two isolated points $(1,1)$ and $(0,0)$ of the double arrow space, make sure that $U_a$ maps these two points to the value 0. The following is a precise definition of the function $U_a$ .

$\displaystyle U_a(y) = \left\{ \begin{array}{ll} \displaystyle 1 &\ \ \ \ \ \ y \in [a,1) \times \{ 1 \} \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ y \in (a,1] \times \{ 0 \} \\ \text{ } & \text{ } \\ 0 &\ \ \ \ \ \ y \in (0,a] \times \{ 0 \} \\ \text{ } & \text{ } \\ 0 &\ \ \ \ \ \ y \in [0,a) \times \{ 1 \} \\ \text{ } & \text{ } \\ 0 &\ \ \ \ \ \ y=(0,0) \text{ or } y = (1,1) \end{array} \right.$

The resulting $U_a$ is a translation of $f_a$ . Under the embedding $E$ defined earlier, we see that $E(U_a)=f_a$ . Let $U=\{ U_a: 0<a<1 \}$ . The set $U$ in $C_p(D)$ is homeomorphic to the set $F$ in $C_p(S)$ . Thus $U$ is a closed and discrete subspace of $C_p(D)$ since $F$ is a closed and discrete subspace of $C_p(S)$ .

Remarks

The drawings and the embedding discussed here and in the previous post establish that $C_p(D)$ , the space of continuous functions on the double arrow space, contains a closed and discrete subspace of cardinality continuum. It follows that $C_p(D)$ is not normal. This is due to the fact that if $C_p(X)$ is normal, then $C_p(X)$ must have countable extent (i.e. all closed and discrete subspaces must be countable).

While $C_p(D)$ is embedded in $C_p(S)$ , the function space $C_p(S)$ is not embedded in $C_p(D)$ . Because the double arrow space is compact, $C_p(D)$ has countable tightness. If $C_p(S)$ were to be embedded in $C_p(D)$ , then $C_p(S)$ would be countably tight too. However, $C_p(S)$ is not countably tight due to the fact that $S \times S$ is not Lindelof (see Theorem 1 in this previous post).

Reference

Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
Tkachuk V. V., A $C_p$ -Theory Problem Book, Topological and Function Spaces, Springer, New York, 2011.

$\text{ }$

Dan Ma math

Daniel Ma mathematics

Drawing Sorgenfrey continuous functions

Posted on September 5, 2017 by Dan Ma

The Sorgenfrey line is a well known topological space. It is the real number line with open intervals defined as sets of the form $[a,b)$ . Though this is a seemingly small tweak, it generates a vastly different space than the usual real number line. In this post, we look at the Sorgenfrey line from the continuous function perspective, in particular, the continuous functions that map the Sorgenfrey line into the real number line. In the process, we obtain insight into the space of continuous functions on the Sorgenfrey line.

The next post is a continuation on the theme of drawing Sorgenfrey continuous functions.

The Sorgenfrey Line

Let $\mathbb{R}$ denote the real number line. The usual open intervals are of the form $(a,b)=\left\{x \in \mathbb{R}: a<x<b \right\}$ . The union of such open intervals is called an open set. If more than one topologies are considered on the real line, these open sets are referred to as the usual open sets or Euclidean open sets (on the real line). The open intervals $(a,b)$ form a base for the usual topology on the real line. One important fact abut the usual open sets is that the usual open sets can be generated by the intervals $(a,b)$ where both end points are rational numbers. Thus the usual topology on the real line is said to have a countable base.

Now tweak the usual topology by calling sets of the form $[a,b)=\left\{x \in \mathbb{R}: a \le x<b \right\}$ open intervals. Then form open sets by taking unions of all such open intervals. The collection of such open sets is called the Sorgenfrey topology (on the real line). The real number line $\mathbb{R}$ with the Sorgenfry topology is called the Sorgenfrey line, denoted by $\mathbb{S}$ . The Sorgenfrey line has been discussed in this blog, starting with this post. This post examines continuous functions from $\mathbb{S}$ into the real line. In the process, we gain insight on the space of continuous functions defined on $\mathbb{S}$ .

Note that any usual open interval $(a,b)$ is the union of intervals of the form $[c,d)$ . Thus any usual (Euclidean) open set is an open set in the Sorgenfrey line. Thus the usual topology (on the real line) is contained in the Sorgenfrey topology, i.e. the usual topology is a weaker (coarser) topology.

Let $C(\mathbb{R})$ be the set of all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ where the domain is the real number line with the usual topology. Let $C(\mathbb{S})$ be the set of all continuous functions $f:\mathbb{S} \rightarrow \mathbb{R}$ where the domain is the Sorgenfrey line. In both cases, the range is always the number line with the usual topology. Based on the preceding paragraph, any continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ is also continuous with respect to the Sorgenfrey line, i.e. $C(\mathbb{R}) \subset C(\mathbb{S})$ .

Pictures of Continuous Functions

Consider the following two continuous functions.

Figure 1 – CDF of the standard normal distribution

Figure 2 – CDF of the uniform distribution

The first one (Figure 1) is the cumulative distribution function (CDF) of the standard normal distribution. The second one (Figure 2) is the CDF of the uniform distribution on the interval $(0,a)$ where $a>0$ . Both of these are continuous in the usual Euclidean topology (in the domain). Such graphs would make regular appearance in a course on probability and statistics. They also show up in a calculus course as an everywhere differentiable curve (Figure 1) and as a differentiable curve except at finitely many points (Figure 2). Both of these functions can also be regarded as continuous functions on the Sorgenfrey line.

Consider a function that is continuous in the Sorgenfrey line but not continuous in the usual topology.

Figure 3 – Right continuous function

Figure 3 is a function that maps the interval $(-\infty,0)$ to -1 and maps the interval $[0,\infty)$ to 1. It is not continuous in the usual topology because of the jump at $x=0$ . But it is a continuous function when the domain is considered to be the Sorgenfrey line. Because of the open intervals being $[a,b)$ , continuous functions defined on the Sorgenfrey line are right continuous.

The cumulative distribution function of a discrete probability distribution is always right continuous, hence continuous in the Sorgenfrey line. Here’s an example.

Figure 4 – CDF of a discrete uniform distribution

Figure 4 is the CDF of the uniform distribution on the finite set $\left\{0,1,2,3,4 \right\}$ , where each point has probability 0.2. There is a jump of height 0.2 at each of the points from 0 to 4. Figure 3 and Figure 4 are step functions. As long as the left point of a step is solid and the right point is hollow, the step functions are continuous on the Sorgenfrey line.

The take away from the last four figures is that the real-valued continuous functions defined on the Sorgenfrey line are right continuous and that step functions (with the left point solid and the right point hollow) are Sorgenfrey continuous.

A Family of Sorgenfrey Continuous Functions

The four examples of continuous functions shown above are excellent examples to illustrate the Sorgenfrey topology. We now introduce a family of continuous functions $f_a:\mathbb{S} \rightarrow \mathbb{R}$ for $0<a<1$ . These continuous functions will lead to additional insight on the function space whose domain space is the Sorgenfrey line.

For any $0<a<1$ , the following gives the definition and the graph of the function $f_a$ .

$\displaystyle f_a(x) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ -\infty<x<-1 \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ -1 \le x<-a \\ \text{ } & \text{ } \\ 0 &\ \ \ \ \ \ -a \le x <a \\ \text{ } & \text{ } \\ 1 &\ \ \ \ \ \ a \le x <1 \\ \text{ } & \text{ } \\ 0 &\ \ \ \ \ \ 1 \le x <\infty \end{array} \right.$

Figure 5 – a family of Sorgenfrey continuous functions

Function Space on the Sorgenfrey Line

This is the place where we switch the focus to function space. The set $C(\mathbb{S})$ is a subset of the product space $\mathbb{R}^\mathbb{R}$ . So we can consider $C(\mathbb{S})$ as a topological space endowed with the topology inherited as a subspace of $\mathbb{R}^\mathbb{R}$ . This topology on $C(\mathbb{S})$ is called the pointwise convergence topology and $C(\mathbb{S})$ with the product subspace topology is denoted by $C_p(\mathbb{S})$ . See here for comments on how to work with the pointwise convergence topology.

For the present discussion, all we need is some notation on a base for $C_p(\mathbb{S})$ . For $x \in \mathbb{S}$ , and for any open interval $(a,b)$ (open in the usual topology of the real number line), let $[x,(a,b)]=\left\{h \in C_p(\mathbb{S}): h(x) \in (a, b) \right\}$ . Then the collection of intersections of finitely many $[x,(a,b)]$ would form a base for $C_p(\mathbb{S})$ .

The following is the main fact we wish to establish.

The function space $C_p(\mathbb{S})$ contains a closed and discrete subspace of cardinality continuum. In particular, the set $F=\left\{f_a: 0<a<1 \right\}$ is a closed and discrete subspace of $C_p(\mathbb{S})$ .

The above result will derive several facts on the function space $C_p(\mathbb{S})$ , which are discussed in a section below. More interestingly, the proof of the fact that $F=\left\{f_a: 0<a<1 \right\}$ is a closed and discrete subspace of $C_p(\mathbb{S})$ is based purely on the definition of the functions $f_a$ and the Sorgenfrey topology. The proof given below does not use any deep or high powered results from function space theory. So it should be a nice exercise on the Sorgenfrey topology.

I invite readers to either verify the fact independently of the proof given here or follow the proof closely. Lots of drawing of the functions $f_a$ on paper will be helpful in going over the proof. In this one instance at least, drawing continuous functions can help gain insight on function spaces.

Working out the Proof

The following diagram was helpful to me as I worked out the different cases in showing the discreteness of the family $F=\left\{f_a: 0<a<1 \right\}$ . The diagram is a valuable aid in convincing myself that a given case is correct.

Figure 6 – A comparison of three Sorgenfrey continuous functions

Now the proof. First, $F$ is relatively discrete in $C_p(\mathbb{S})$ . We show that for each $a$ , there is an open set $O$ containing $f_a$ such that $O$ does not contain $f_w$ for any $w \ne a$ . To this end, let $O=[a,V_1] \cap [-a,V_2]$ where $V_1$ and $V_2$ are the open intervals $V_1=(0.9,1.1)$ and $V_2=(-0.1,0.1)$ . With Figure 6 as an aid, it follows that for $0<b<a$ , $f_b \notin O$ and for $a<c<1$ , $f_c \notin O$ .

The open set $O=[a,V_1] \cap [-a,V_2]$ contains $f_a$ , the function in the middle of Figure 6. Note that for $0<b<a$ , $f_b(-a)=1$ and $f_b(-a) \notin V_2$ . Thus $f_b \notin O$ . On the other hand, for $a<c<1$ , $f_c(a)=0$ and $f_c(a) \notin V_1$ . Thus $f_c \notin O$ . This proves that the set $F$ is a discrete subspace of $C_p(\mathbb{S})$ relative to $F$ itself.

Now we show that $F$ is closed in $C_p(\mathbb{S})$ . To this end, we show that

$g \in C_p(\mathbb{S})$

$U$

$g$

$U$

$F$

Actually, this has already been done above with points $g$ that are in $F$ . One thing to point out is that the range of $f_a$ is $\left\{0,1 \right\}$ . As we consider $g \in C_p(\mathbb{S})$ , we only need to consider $g$ that maps into $\left\{0,1 \right\}$ . Let $g \in C_p(\mathbb{S})$ . The argument is given in two cases regarding the function $g$ .

Case 1. There exists some $a \in (0,1)$ such that $g(a) \ne g(-a)$ .

We assume that $g(a)=0$ and $g(-a)=1$ . Then for all $0<b<a$ , $f_b(a)=1$ and for all $a<c<1$ , $f_c(-a)=0$ . Let $U=[a,(-0.1,0.1)] \cap [-a,(0.9,1.1)]$ . Then $g \in U$ and $U$ contains no $f_b$ for any $0<b<a$ and $f_c$ for any $a<c<1$ . To help see this argument, use Figure 6 as a guide. The case that $g(a)=1$ and $g(-a)=0$ has a similar argument.

Case 2. For every $a \in (0,1)$ , we have $g(a) = g(-a)$ .

Claim. The function $g$ is constant on the interval $(-1,1)$ . Suppose not. Let $0<b<a<1$ such that $g(a) \ne g(b)$ . Suppose that $0=g(b) < g(a)=1$ . Consider $W=\left\{w<a: g(w)=0 \right\}$ . Clearly the number $a$ is an upper bound of $W$ . Let $u \le a$ be a least upper bound of $W$ . The function $g$ has value 1 on the interval $(u,a)$ . Otherwise, $u$ would not be the least upper bound of the set $W$ . There is a sequence of points $\left\{x_n \right\}$ in the interval $(b,u)$ such that $x_n \rightarrow u$ from the left such that $g(x_n)=0$ for all $n$ . Otherwise, $u$ would not be the least upper bound of the set $W$ .

It follows that $g(u)=1$ . Otherwise, the function $g$ is not continuous at $u$ . Now consider the 6 points $-a<-u<-b<b<u<a$ . By the assumption in Case 2, $g(u)=g(-u)=1$ and $g(b)=g(-b)=0$ . Since $g(x_n)=0$ for all $n$ , $g(-x_n)=0$ for all $n$ . Note that $-x_n \rightarrow -u$ from the right. Since $g$ is right continuous, $g(-u)=0$ , contradicting $g(-u)=1$ . Thus we cannot have $0=g(b) < g(a)=1$ .

Now suppose we have $1=g(b) > g(a)=0$ where $0<b<a<1$ . Consider $W=\left\{w<a: g(w)=1 \right\}$ . Clearly $W$ has an upper bound, namely the number $a$ . Let $u \le a$ be a least upper bound of $W$ . The function $g$ has value 0 on the interval $(u,a)$ . Otherwise, $u$ would not be the least upper bound of the set $W$ . There is a sequence of points $\left\{x_n \right\}$ in the interval $(b,u)$ such that $x_n \rightarrow u$ from the left such that $g(x_n)=1$ for all $n$ . Otherwise, $u$ would not be the least upper bound of the set $W$ .

It follows that $g(u)=0$ . Otherwise, the function $g$ is not continuous at $u$ . Now consider the 6 points $-a<-u<-b<b<u<a$ . By the assumption in Case 2, $g(u)=g(-u)=0$ and $g(b)=g(-b)=1$ . Since $g(x_n)=1$ for all $n$ , $g(-x_n)=1$ for all $n$ . Note that $-x_n \rightarrow -u$ from the right. Since $g$ is right continuous, $g(-u)=1$ , contradicting $g(-u)=0$ . Thus we cannot have $1=g(b) > g(a)=0$ .

The claim that the function $g$ is constant on the interval $(-1,1)$ is established. To wrap up, first assume that the function $g$ is 1 on the interval $(-1,1)$ . Let $U=[0,(0.9,1.1)]$ . It is clear that $g \in U$ . It is also clear from Figure 5 that $U$ contains no $f_a$ . Now assume that the function $g$ is 0 on the interval $(-1,1)$ . Since $g$ is Sorgenfrey continuous, it follows that $g(-1)=0$ . Let $U=[-1,(-0.1,0.1)]$ . It is clear that $g \in U$ . It is also clear from Figure 5 that $U$ contains no $f_a$ .

We have established that the set $F=\left\{f_a: 0<a<1 \right\}$ is a closed and discrete subspace of $C_p(\mathbb{S})$ .

What does it Mean?

The above argument shows that the set $F$ is a closed an discrete subspace of the function space $C_p(\mathbb{S})$ . We have the following three facts.

Three Results

$C_p(\mathbb{S})$ is separable.
$C_p(\mathbb{S})$ is not hereditarily separable.
$C_p(\mathbb{S})$ is not a normal space.

To show that $C_p(\mathbb{S})$ is separable, let’s look at one basic helpful fact on $C_p(X)$ . If $X$ is a separable metric space, e.g. $X=\mathbb{R}$ , then $C_p(X)$ has quite a few nice properties (discussed here). One is that $C_p(X)$ is hereditarily separable. Thus $C_p(\mathbb{R})$ , the space of real-valued continuous functions defined on the number line with the pointwise convergence topology, is hereditarily separable and thus separable. Recall that continuous functions in $C_p(\mathbb{R})$ are also Soregenfrey line continuous. Thus $C_p(\mathbb{R})$ is a subspace of $C_p(\mathbb{S})$ . The space $C_p(\mathbb{R})$ is also a dense subspace of $C_p(\mathbb{S})$ . Thus the space $C_p(\mathbb{S})$ contains a dense separable subspace. It means that $C_p(\mathbb{S})$ is separable.

Secondly, $C_p(\mathbb{S})$ is not hereditarily separable since the subspace $F=\left\{f_a: 0<a<1 \right\}$ is a closed and discrete subspace.

Thirdly, $C_p(\mathbb{S})$ is not a normal space. According to Jones’ lemma, any separable space with a closed and discrete subspace of cardinality of continuum is not a normal space (see Corollary 1 here). The subspace $F=\left\{f_a: 0<a<1 \right\}$ is a closed and discrete subspace of the separable space $C_p(\mathbb{S})$ . Thus $C_p(\mathbb{S})$ is not normal.

Remarks

The topology of the Sorgenfrey line is vastly different from the usual topology on the real line even though the the Sorgenfrey topology is obtained by a seemingly small tweak from the usual topology. The real line is a metric space while the Sorgenfrey line is not metrizable. The real number line is connected while the Sorgenfrey line is not. The countable power of the real number line is a metric space and thus a normal space. On the other hand, the Sorgenfrey line is a classic example of a normal space whose square is not normal. See here for a basic discussion of the Sorgenfrey line.

The pictures of Sorgenfrey continuous functions demonstrated here show that the real number line and the Sorgenfrey line are also very different from a function space perspective. The function space $C_p(\mathbb{R})$ has a whole host of nice properties: normal, Lindelof (hence paracompact and collectionwise normal), hereditarily Lindelof (hence hereditarily normal), hereditarily separable, and perfectly normal (discussed here).

Though separable, the function space $C_p(\mathbb{S})$ contains a closed and discrete subspace of cardinality continuum, making it not hereditarily separable and not normal.

For more information about $C_p(X)$ in general and $C_p(\mathbb{S})$ in particular, see [1] and [2]. A different proof that $C_p(\mathbb{S})$ contains a closed and discrete subspace of cardinality continuum can be found in Problem 165 in [2].

The next post is a continuation on the theme of drawing Sorgenfrey continuous functions.

Reference

Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
Tkachuk V. V., A $C_p$ -Theory Problem Book, Topological and Function Spaces, Springer, New York, 2011.

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