A Space with G-delta Diagonal that is not Submetrizable

The property of being submetrizable implies having a G_\delta-diagonal. There are several other properties lying between these two properties (see [1]). Before diving into these other properties, it may be helpful to investigate a classic example of a space with a G_\delta-diagonal that is not submetrizable.

The diagonal of a space X is the set \Delta=\left\{(x,x): x \in X \right\}, a subset of the square X \times X. An interesting property is when the diagonal of a space is a G_\delta-set in X \times X (the space is said to have a G_\delta-diagonal). Any compact space or a countably compact space with this property must be metrizable (see compact and countably compact space). A space (X,\tau) is said to be submetrizable if there is a topology \tau^* that can be defined on X such that (X,\tau^*) is a metrizable space and \tau^* \subset \tau. In other words, a submetrizable space is a space that has a coarser (weaker) metrizable topology. Every submetrizable space has a G_\delta-diagonal. Note that when X has a weaker metric topology, the diagonal \Delta is always a G_\delta-set in the metric square X \times X (hence in the square in the original topology). The property of having a G_\delta-diagonal is strictly weaker than the property of having a weaker metric topology. In this post, we discuss the Mrowka space, which is a classic example of a space with a G_\delta-diagonal that is not submetrizable.

The Mrowka space (also called Psi space) was discussed previously in this blog (see this post). For the sake of completeness, the example is defined here.

First, we define some basic notions. Let \omega be the first infinite ordinal (or more conveniently the set of all nonnegative integers). Let \mathcal{A} be a family of infinite subsets of \omega. The family \mathcal{A} is said to be an almost disjoint family if for each two distinct A,B \in \mathcal{A}, A \cap B is finite. An almost disjoint family \mathcal{A} is said to be a maximal almost disjoint family if B is an infinite subset of \omega such that B \notin \mathcal{A}, then B \cap A is infinite for some A \in \mathcal{A}. In other words, if you put one more set into a maximal almost disjoint family, it ceases to be almost disjoint.

A natural question is whether there is an uncountable almost disjoint family of subsets of \omega. In fact, there is one whose cardinality is continuum (the cardinality of the real line). To see this, identify \omega with \mathbb{Q}=\lbrace{r_0,r_1,r_2,...}\rbrace (the set of all rational numbers). Let \mathbb{P}=\mathbb{R}-\mathbb{Q} be the set of all irrational numbers. For each x \in \mathbb{P}, choose a subsequence of \mathbb{Q} consisting of distinct elements that converges to x (in the Euclidean topology). Then the family of all such sequences of rational numbers would be an almost disjoint family. By a Zorn’s Lemma argument, this almost disjoint family is contained within a maximal almost disjoint family. Thus we also have a maximal almost disjoint family of cardinality continuum. On the other hand, there is no countably infinite maximal almost disjoint family of subsets of \omega (see this post).

Let \mathcal{A} be an infinite almost disjoint family of subsets of \omega. We now define a Mrowka space (or \Psi-space), denoted by \Psi(\mathcal{A}). The underlying set is \Psi(\mathcal{A})=\mathcal{A} \cup \omega. Points in \omega are isolated. For A \in \mathcal{A}, a basic open set is of the form \lbrace{A}\rbrace \cup (A-F) where F \subset \omega is finite. It is straightforward to verify that \Psi(\mathcal{A}) is Hausdorff, first countable and locally compact. It has a countable dense set of isolated points. Note that \mathcal{A} is an infinite discrete and closed set in the space \Psi(\mathcal{A}). Thus \Psi(\mathcal{A}) is not countably compact.

We would like to point out that the definition of a Mrowka space \Psi(\mathcal{A}) only requires that the family \mathcal{A} is an almost disjoint family and does not necessarily have to be maximal. For the example discribed in the title, \mathcal{A} needs to be a maximal almost disjoint family of subsets of \omega.

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Example
Let \mathcal{A} be a maximal almost disjoint family of subsets of \omega. Then \Psi(\mathcal{A}) as defined above is a space in which there is a G_\delta-diagonal that is not submetrizable.

Note that \Psi(\mathcal{A}) is pseudocompact (proved in this post). Because there is no countable maximal almost disjoint family of subsets of \omega, \mathcal{A} must be an uncountable in addition to being a closed and discrete subspace of \Psi(\mathcal{A}) (thus the space is not Lindelof). Since \Psi(\mathcal{A}) is separable and is not Lindelof, \Psi(\mathcal{A}) is not metrizable. Any psuedocompact submetrizable space is metrizable (see Theorem 4 in this post). Thus \Psi(\mathcal{A}) must not be submetrizable.

On the other hand, any \Psi-space \Psi(\mathcal{A}) (even if \mathcal{A} is not maximal) is a Moore space. It is well known that any Moore space has a G_\delta-diagonal. The remainder of this post has a brief discussion of Moore space.

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Moore Space

A sequence \lbrace{\mathcal{D}_n}\rbrace_{n<\omega} of open covers of a space X is a development for X if for each x \in X and each open set U \subset X with x \in U, there is some n such that any open set in \mathcal{D}_n containing the point x is contained in U. A developable space is one that has a development. A Moore space is a regular developable space.

Suppose that X is a Moore space. We show that X has a G_\delta-diagonal. That is, we wish to show that \Delta=\left\{(x,x): x \in X \right\} is a G_\delta-set in X \times X.

Let \lbrace{\mathcal{D}_n}\rbrace_{n<\omega} be a development. For each n, let U_n=\bigcup \lbrace{V \times V:V \in \mathcal{D}_n}\rbrace. Clearly \Delta \subset \bigcap_{n<\omega} U_n. Let (x,y) \in \bigcap_{n<\omega} U_n. For each n, (x,y) \in V_n \times V_n for some V_n \in \mathcal{D}_n. We claim that x=y. Suppose that x \ne y. By the definition of development, there exists some m such that every open set in \mathcal{D}_m containing the point x has to be a subset of X-\left\{y \right\}. Then V_m \subset X-\left\{y \right\}, which contradicts y \in V_m. Thus we have \Delta = \bigcap_{n<\omega} U_n.

The remaining thing to show is that \Psi(\mathcal{A}) is a Moore space. For each positive integer n, let F_n=\left\{0,1,\cdots,n-1 \right\} and let F_0=\varnothing. The development is defined by \lbrace{\mathcal{E}_n}\rbrace_{n<\omega}, where for each n, \mathcal{E}_n consists of open sets of the form \lbrace{A}\rbrace \cup (A-F_n) where A \in \mathcal{A} plus any singleton \left\{j \right\} (j \in \omega) that has not been covered by the sets \lbrace{A}\rbrace \cup (A-F_n).

Reference

  1. Arhangel’skii, A. V., Buzyakova, R. Z., The rank of the diagonal and submetrizability, Commentationes Mathematicae Universitatis Carolinae, Vol. 47 (2006), No. 4, 585-597.
  2. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  3. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

When is a Pseudocompact Space Metrizable?

Compactness, countably compactness and pseudocompactness are three successively weaker properties. It follows easily from definitions that

(A) \ \ \ \ \ \text{compact} \Rightarrow \text{countably compact} \Rightarrow \text{pseudocompact}

None of these arrows can be reversed. It is well known that either compactness or countably complactness plus having a G_\delta-diagonal implies metrizability. We have:

(B) \ \ \ \ \ \text{compact} + \text{having a } G_\delta \text{-diagonal} \Rightarrow \text{metrizable}

(C) \ \ \ \ \ \text{countably compact} + \text{having a } G_\delta \text{-diagonal} \Rightarrow \text{metrizable}

A question can be asked whether these results can be extended to pseudocompactness.

Question (D) \ \ \ \ \ \text{pseudocompact compact} + \text{having a } G_\delta \text{-diagonal} \Rightarrow \text{metrizable?}

The answer to this question is no. The space defined using a maximal almost disjoint family of subsets of \omega is an example of a non-metrizable pseudocompact space with a G_\delta-diagonal (discussed in this post). In this post we show that if we strengthen “having a G_\delta-diagonal” to being submetrizable, we have a theorem. Specifically, we show:

(E) \ \ \ \ \ \text{pseudocompact} + \text{submetrizable} \Rightarrow \text{metrizable}

For the result of (B), see this post. For the result of (C), see this post. In this post, we discuss the basic properties of pseudocompactness that build up to the result of (E). All spaces considered here are at least Tychonoff (i.e. completely regular). For any basic notions not defined here, see [1] or [2].

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Pseudocompact Spaces

A space X is said to be pseudocompact if every real-valued continuous function defined on X is a bounded function. Any real-valued continuous function defined on a compact space must be bounded (and is thus pseudocomppact). If there were an unbounded real-valued continuous function defined on a space X, then X would have a countably infinite discrete set (thus not countably compact). Thus countably compact implies pseudocompact, as indicated by (A).

A space X is submetrizable if there is a coarser (i.e. weaker) topology that is a metrizable topology. Specifically the topological space (X,\tau) is submetrizable if there is another topology \tau^* that can be defined on X such that \tau^* \subset \tau and (X,\tau^*) is metrizable. The Sorgenfrey line is non-metrizable and yet the Sorgenfrey topology has a weaker topology that is metrizable, namely the Euclidean topology of the real line.

The following two theorems characterizes pseudocompact spaces in terms of locally finite open family of open sets (Theorem 1) and the finite intersection property (Theorem 2). Both theorems are found in Engelking (Theorem 3.10.22 and Theorem 3.10.23 in page 207 of [1]). Theorem 3 states that in a pseudocompact space, closed domains are pseudocompact (the definition of closed domain is stated before the theorem). Theorem 4 is the main theorem (result E stated above).

Theorem 1
Let X be a space. The following conditions are equivalent:

  1. The space X is pseudocompact.
  2. If \mathcal{V} is a locally finite family of non-empty open subsets of X, then \mathcal{V} is finite.
  3. If \mathcal{V} is a locally finite open cover of X, then \mathcal{V} is finite.
  4. If \mathcal{V} is a locally finite open cover of X, then \mathcal{V} has a finite subcover.

Proof
1 \Rightarrow 2
Suppose that condition 2 does not hold. Then there is an infinite locally finite family of non-empty open sets \mathcal{V} such that \mathcal{V}=\left\{V_1,V_2,V_3,\cdots \right\}. We wish to define an unbounded continuous function using \mathcal{V}.

This is where we need to invoke the assumption of complete regularity. For each n choose a point x_n \in V_n. Then for each n, there is a continuous function f_n:X \rightarrow [0,n] such that f_n(x_n)=n and f_n(X-V_n) \subset \left\{ 0 \right\}. Define f:X \rightarrow [0,\infty) by f(x)=f_1(x)+f_2(x)+f_3(x)+\cdots.

Because \mathcal{V} is locally finite, the function f is essentially pointwise the sum of finitely many f_n. In other words, for each x \in X, for some positive integer N, f_j(x)=0 for all j \ge N. Thus the function f is well defined and is continuous at each x \in X. Note that for each x_n, f(x_n) \ge n, showing that it is unbounded.

The directions 2 \Rightarrow 3 and 3 \Rightarrow 4 are clear.

4 \Rightarrow 1
Let g:X \rightarrow \mathbb{R} be a continuous function. We want to show that g is a bounded function. Consider the open family \mathcal{O}=\left\{\cdots,O_{-3},O_{-2},O_{-1},O_0,O_1,O_2,O_3,\cdots \right\} where each O_n=g^{-1}((n,n+2)). Note that \mathcal{O} is a locally finite family in X since its members O_n=g^{-1}((n,n+2)) are inverse images of members of a locally finite family in the range space \mathbb{R}. By condition 4, \mathcal{O} has a finite subcover, leading to the conclusion that g is a bounded function. \blacksquare

Theorem 2
Let X be a space. The following conditions are equivalent:

  1. The space X is pseudocompact.
  2. If \mathcal{O}=\left\{O_1,O_2,O_3,\cdots \right\} is a family of non-empty open subsets of X such that O_n \supset O_{n+1} for each n, then \bigcap \limits_{n=1}^\infty \overline{O_n} \ne \varnothing.
  3. If \mathcal{V}=\left\{V_1,V_2,V_3,\cdots \right\} is a family of non-empty open subsets of X such that \mathcal{V} has the finite intersection property, then \bigcap \limits_{n=1}^\infty \overline{V_n} \ne \varnothing.

Proof
1 \Rightarrow 2
Suppose that X is pseudocompact. Suppose \mathcal{O}=\left\{O_1,O_2,O_3,\cdots \right\} satisfies the hypothesis of condition 2. If there is some positive integer m such that O_n=O_m for all n \ge m, then we are done. So assume that O_n are distinct for infinitely many n. According to condition 2 in Theorem 1, \mathcal{O} must not be a locally finite family. Then there exists a point x \in X such that every open set containing x must meet infinitely many O_n. This implies that x \in \overline{O_n} for infinitely many n. Thus x \in \bigcap \limits_{n=1}^\infty \overline{O_n}.

2 \Rightarrow 3
Suppose \mathcal{V}=\left\{V_1,V_2,V_3,\cdots \right\} is a family of non-empty open sets with the finite intersection property as in the hypothesis of 3. Then let O_1=V_1, O_2=V_1 \cap V_2, O_3=V_1 \cap V_2 \cap V_3, and so on. By condition 2, we have \bigcap \limits_{n=1}^\infty \overline{O_n} \ne \varnothing, which implies \bigcap \limits_{n=1}^\infty \overline{V_n} \ne \varnothing.

3 \Rightarrow 1
Let g:X \rightarrow \mathbb{R} be a continuous function such that g is unbounded. For each positive integer n, let V_n=\left\{x \in X: \lvert g(x) \lvert > n \right\}. Clearly the open sets V_n have the finite intersection property. Because g is unbounded, it follows that \bigcap \limits_{n=1}^\infty \overline{V_n} = \varnothing. \blacksquare

Let X be a space. Let A \subset X. The interior of A, denoted by \text{int}(A), is the set of all points x \in X such that there exists an open set O with x \in O \subset A. Points of \text{int}(A) are called the interior points of A. A subset C \subset X is said to be a closed domain if C=\overline{\text{int}(C)}. It is clear that C is a closed domain if and only if C is the closure of an open set.

Theorem 3
The property of being a pseudocompact space is hereditary with respect to subsets that are closed domains.

Proof
Let X be a pseudocompact space. We show that \overline{U} is pseudocompact for any nonempty open set U \subset X. Let Y=\overline{U} where U is a non-empty open subset of X. Let S_1 \supset S_2 \supset S_3 \supset \cdots be a decreasing sequence of open subsets of Y. Note that each S_i contains points of the open set U. Let O_i=S_i \cap U for each i. Note that the open sets O_i form a decreasing sequence of open sets in the pseudocompact space X. By Theorem 2, we have \bigcap \limits_{n=1}^\infty \overline{O_n} \ne \varnothing (closure here is with respect to X). Note that points in \bigcap \limits_{n=1}^\infty \overline{O_n} are also points in \bigcap \limits_{n=1}^\infty \overline{S_n} (closure with respect to Y). By Theorem 2, Y=\overline{U} is pseudocompact. \blacksquare

Theorem 4 (Statement E above)
Let X be a pseudocompact submetrizable space. Then X is metrizable.

Proof
Let (X,\tau) be a pseudocompact submetrizable space. Then there exists topology \tau^* on X such (X,\tau^*) is metrizable and \tau^* \subset \tau. We show that \tau \subset \tau^*, leading to the conclusion that (X,\tau) is also metrizable. If A \subset X, we denote the closure of A in (X,\tau) by cl_{\tau}(A) and the closure of A in (X,\tau^*) by cl_{\tau^*}(A).

To show that \tau \subset \tau^*, we show any closed set with respect to the topology \tau is also a closed set with respect to the topology \tau^*. Let C be a closed set in (X,\tau). Consider the family \mathcal{W}=\left\{cl_{\tau}(U): U \in \tau \text{ and } C \subset U \right\}. We make the following claims.

Claim 1. C=\bigcap \left\{W: W \in \mathcal{W} \right\}.

Claim 2. Each W \in \mathcal{W} is pseudocompact in (X,\tau).

Claim 3. Each W \in \mathcal{W} is pseudocompact in (X,\tau^*).

Claim 4. Each W \in \mathcal{W} is compact in (X,\tau^*).

We now discuss each of these four claims. For Claim 1, it is clear that C \subset \bigcap \left\{W: W \in \mathcal{W} \right\}. The reverse set inclusion follows from the fact that X is a regular space. Claim 2 follows from Theorem 3. Note that sets in \mathcal{W} are closed domains in the pseudocompact space (X,\tau).

If sets in \mathcal{W} are pseudocompact in the larger topology \tau, they would be pseudocompact in the weaker topology \tau^* too. Thus Claim 3 is established. In a metrizable space, compactness and weaker notions such as countably compactness and pseudocompactness coincide. Because they are pseudocompact subsets, sets in \mathcal{W} are compact in the metrizable space (X,\tau^*). Thus Claim 4 is established.

It follows that C is closed in (X,\tau^*) since it is the intersection of compact sets in (X,\tau^*). Thus (X,\tau) is identical to (X,\tau^*), implying that (X,\tau) is metrizable. \blacksquare

Reference

  1. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

Thinking about the Space of Irrationals Topologically

Let \mathbb{R} denote the real number line and \mathbb{P} denote the set of all irrational numbers. The irrational numbers and the set \mathbb{P} occupy an important place in mathematics. The set \mathbb{P} with the Euclidean topology inherited from the full real line is a topological space in its own right. Thus the space \mathbb{P} has some of the same properties inherited from the Euclidean real line, e.g., just to name a few, it is hereditarily separable, hereditarily Lindelof, paracompact and metrizable. The space of the irrational numbers \mathbb{P} has many properties apart from the full real line (e.g. \mathbb{P} is totally disconnected). In this post, we look at a topological description of the space \mathbb{P}.

Let \omega be the set of all nonnegative integers. Then the space \mathbb{P} of irrational numbers is topologically equivalent (i.e. homeomorphic to) the product space \prod \limits_{i=0}^\infty X_i where each X_i=\omega has the discrete topology. We will also denote the product space \prod \limits_{i=0}^\infty X_i by \omega^\omega. We have the following theorem.

Theorem
The space \mathbb{P} of irrational numbers is homeomorphic to the product space \omega^\omega.

Because of this theorem, we can look at irrational numbers as sequences of nonnegative integers. Specifically each irrational number can be identified by a unique sequence of nonnegative integers. We can think of each such unique sequence as an address to locate an irrational number. In the remainder of the post, we describe at a high level how to define the unique addresses, which will also give us a homeomorphic map between the space \mathbb{P} and the product space \omega^\omega.

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Step 0
Put the rational numbers in a sequence r_0,r_1,r_2,r_3,\cdots such that r_0=0. We divide the real line into countably many non-overlapping intervals. Specifically, let A_0=[0,1], A_1=[-1,0], A_2=[1,2], etc (see the following figure).

To facilitate the remaining construction, we denote the left endpoint of the interval A_i by L_i and denote the right endpoint by R_i.

Step 1
In each of the interval A_i, we further divide it into non-overlapping intervals. The left endpoint and the right endpoint of the interval A_i are L_i and R_i, respectively. We choose a sequence x_{i,0}, x_{i,1}, x_{i,2},\cdots of rational numbers converging to the right endpoint R_i. Then let A_{i,0}=[L_i,x_{i,0}], A_{i,1}=[x_{i,0},x_{i,1}], A_{i,2}=[x_{i,1},x_{i,2}], etc (see the following figure).

Two important points to consider in Step 1. One is that we make sure the rational number r_1 is chosen as an endpoint of some interval in Step 1. The second is that the length of each A_{i,j} is less than \displaystyle \frac{1}{2^1}.

Step 2
In each of the interval A_{i,j}, we further divide it into non-overlapping intervals. The left endpoint and the right endpoint of the interval A_{i,j} are L_{i,j} and R_{i,j}, respectively. We choose a sequence x_{i,j,0}, x_{i,j,1}, x_{i,j,2},\cdots of rational numbers converging to the left endpoint L_{i,j}. Then let A_{i,j,0}=[x_{i,j,0},R_{i,j}], A_{i,j,1}=[x_{i,j,1},x_{i,j,0}], A_{i,j,2}=[x_{i,j,2},x_{i,j,1}], etc (see the following figure).

As in the previous step, two important points to consider in Step 2. One is that we make sure the rational number r_2 is chosen as an endpoint of some interval in Step 2. The second is that the length of each A_{i,j,k} is less than \displaystyle \frac{1}{2^2}.

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Remark

In the process described above, the endpoints of the intervals A_{f(0),\cdots,f(n)} are rational numbers and we make sure that all the rational numbers are used as endpoints. We also make sure that the intervals from the successive steps are nested closed intervals with lengths \rightarrow 0. The consequence of this point is that the nested decreasing closed intervals will collapse to one single point (since the real line is a complete metric space) and this single point must be an irrational number (since all the rational numbers are used up as endpoints of the nested closed intervals).

In Step i where i is an odd integer, we make the endpoints of the new intervals converge to the right. In Step i where i>1 is an even integer, we make the endpoints of the new intervals converge to the left. This manipulation is to ensure that the nested closed intervals will never share the same endpoint from one step all the way to the end of the process.

Thus if we have f \in \omega^\omega, then \bigcap \limits_{n=0}^\infty A_{f(0),\cdots,f(n)} \ne \varnothing and in fact has only one point that is an irrational number.

On the other hand, for each point x \in \mathbb{P}, we can locate inductively a sequence of intervals, A_{f(0)}, A_{f(0),f(1)}, A_{f(0),f(1),f(2)}, \cdots, containing the point x. This sequence of nested closed intervals must collapse to a single point and the single point must be the irrational number x.

The process described above gives us a one-to-one mapping from \mathbb{P} onto the product space \omega^\omega. This mapping is also continuous in both directions, making it a homeomorphism. the nested intervals defined above form a base for the Euclidean topology on \mathbb{P}. These basic open sets have a natural correspondance with basic open sets in the product space \omega^\omega.

For example, for f \in \omega^\omega, \left\{ A_{f(0),\cdots,f(n)} \cap \mathbb{P}: n \in \omega \right\} is a local base at the point x \in \bigcap \limits_{n=0}^\infty A_{f(0),\cdots,f(n)}. One the other hand, each A_{f(0),\cdots,f(n)} \cap \mathbb{P} has a natural counterpart in a basic open set in the product space, namely the following set:

\displaystyle . \ \ \ \ \left\{g \in \omega^\omega: g \restriction n = f \restriction n \right\}

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The above process establishes that the countable product of the integers, \omega^\omega, is equivalent topologically to the Euclidean space \mathbb{P}.

Elementary Examples of Lindelof Spaces and Separable Spaces

The Euclidean spaces \mathbb{R} and \mathbb{R}^n are both Lindelof and separable. In fact these two properties are equivalent in the class of metrizable spaces. A space is metrizable if its topology can be induced by a metric. In a metrizable space, having one of these properties implies the other one. Any students in beginning topology courses who study basic notions such as the Lindelof property and separability must venture outside the confine of Euclidean spaces or metric spaces. The goal of this post is to present some elementary examples showing that these two notions are not equivalent.

All topological spaces under consideration are Hausdorff. Let X be a space. Let D \subset X. The set D is said to be dense in X if every nonempty open subset of X contains some point of D. The space X is said to be separable if there is countable subset of X that is also dense in X. All Euclidean spaces are separable. For example, in the real line \mathbb{R}, every open interval contains a rational number. Thus the set of all rational numbers \mathbb{Q} is dense in \mathbb{R}.

Let \mathcal{U} be a collection of subsets of the space X. The collection \mathcal{U} is said to be a cover of X if every point of X is contained in some element of \mathcal{U}. The collection \mathcal{U} is said to be an open cover of X if, in addition it being a cover, \mathcal{U} consists of open sets in X.

Let \mathcal{U} be a cover of the space X. Let \mathcal{V} \subset \mathcal{U}. If the collection \mathcal{V} is also a cover of X, we say that \mathcal{V} is a subcover of \mathcal{U}. The space X is a Lindelof space (or has the Lindelof property) if every open cover of X has a countable subcover.

The real \mathbb{R} is Lindelof. Both the Lindelof property and the separability of \mathbb{R} follows from the fact that the Euclidean topology on \mathbb{R} can be generated by a countable base (e.g. one countable base consists of all open intervals with rational endpoints). Now some non-Euclidean (and non-metrizable) examples.

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Example 1 – A Lindelof space that is not separable
Let X be any uncountable set. Let p be a point that is not in X, e.g., let p=\left\{ X \right\}. Define the space Y = \left\{p\right\} \cup X as follows. Let every point in X be isolated, meaning any singleton set \left\{ x \right\} is declared open for any x \in X. An open neighborhood of the point p is of the form \left\{p\right\} \cup W where X-W is a countable subset of X.

It is clear that the resulting space Y is Lindelof since every open set containing p contains all but countably many points of X. It is also clear that no countable set can be dense in Y.

Even though this example Y is Lindelof, it is not hereditarily Lindelof since the subspace X is uncountable discrete space.

In a previous post, we showed that the space Y defined in this example is a productively Lindelof space (meaning that its product with every Lindelof space is Lindelof).

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Remark

A space is said to have the countable chain condition (CCC) if there are no uncountable family of pairwise disjoint open subsets. It is clear that any separable space has the CCC. It follows that the space Y in Example 1 does not have the CCC, since the singleton sets \left\{ x \right\} (with x \in X) forms a pairwise disjoint collection of open sets, showing that the Lindelof property does not even imply the weaker property of having the CCC.

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Example 2 – A separable space that is not Lindelof
The example here is the Tangent Disc Space (Niemytzki’s Tangent Disc Topology in [2]). The underlying set is the upper half plane (the x-axis and the plane above the x-axis). In other words, consider the following set:

\displaystyle . \ \ \ \ \ X=\left\{(x,y) \in \mathbb{R}^2: y \ge 0 \right\}

Let \displaystyle X_u=\left\{(x,y) \in \mathbb{R}^2: y>0 \right\} and T=\left\{(x,0): x \in \mathbb{R} \right\}. The line T is the x-axis and X_u is the upper plane without the x-axis. We define a topology on X such that X_u as a subspace in this topology is Euclidean. The open neighborhoods of a point p=(x,0) \in T are of the form \left\{p \right\} \cup D where D is an open disc tangent to the x-axis at the point p. The figure below illustrates how open neighborhoods at the x-axis are defined.

It is clear that the points with rational coordinates in the upper half plane X_u form a dense set in the tangent disc topology. Thus X is separable. In any Lindelof space, there are no uncountable closed and discrete subsets. Note that the x-axis T is a closed and discrete subspace in the tangent disc space. Thus X is not Lindelof.

Though separable, the Tangent Disc Space is not hereditarily separable since the x-axis T is uncountable and discrete.

The Tangent Disc Space is an interesting example. For example, it is a completely regular space that is an example of a Moore space that is not normal. For these and other interesting facts about the Tangent Disc Space, see [2].

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For the Lindelof property and the property of being separable, there are plenty of examples of spaces that possess only one of the properties. All three references indicated below are excellent places to look. The book by Steen and Seebach ([2]) is an excellent catalog of interesting spaces (many of them are elementary).
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Reference

  1. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Steen, L. A., Seebach, J. A.,Counterexamples in Topology, 1995, Dover Edition, Dover Publications, New York.
  3. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

The product of first countable spaces

All spaces under consideration are Hausdorff. First countable spaces are those spaces where there is a countable local base at every point in the space. This is quite a strong property. For example, every first countable space that is also compact has a cap on its cardinality and the cap is the cardinality of the real line (the continuum). See The cardinality of compact first countable spaces, I in this blog. In fact, if the compact and first countable space is uncountable, it has cardinality continuum (see The cardinality of compact first countable spaces, III). Any metric space (or metrizable space) is first countable. In this post, we discuss the product of first countable spaces. In this regard, first countable spaces and metrizable spaces behave similarly. We show that the product of countably many first countable spaces is first countable while the product of uncountably many first countable is not first countable. For more information on the product topology, see [2].

The Product Space
Consider a collection of sets A_\alpha where \alpha \in S. Let W=\bigcup \limits_{\alpha \in S} A_\alpha. The product \prod \limits_{\alpha \in S} A_\alpha is the set of all functions f:S \mapsto W such that for each \alpha \in S, f(\alpha) \in A_\alpha. If the index set S=\left\{1,2,\cdots,n\right\} is finite, the functions f can be regarded as sequences (f_1,f_2,\cdots,f_n) where each f_i \in A_i. If the index set S=\mathbb{N}, we can think of elements f of the product as the sequence (f_1,f_2,\cdots) where each f_i \in A_i. In general we can regard f \in \prod \limits_{\alpha \in S} A_\alpha as functions f:S \mapsto W or as sequences f=(f_\alpha)_{\alpha \in S}.

Consider the topological spaces X_\alpha where \alpha \in S. Let X=\prod \limits_{\alpha \in S} X_\alpha be the product as defined above. The product space of the spaces X_\alpha is X with the topology defined in the following paragraph.

Let \tau_\alpha be the topology of each space X_\alpha, \alpha \in S. Consider Y=\prod \limits_{\alpha \in S} O_\alpha where for each \alpha \in S, O_\alpha \in \tau_\alpha (i.e. O_\alpha is open in X_\alpha) and O_\alpha=X_\alpha for all but finitely many \alpha \in S. The set of all such sets Y is a base for a topology on the product X=\prod \limits_{\alpha \in S} X_\alpha. This topology is called the product topology of the spaces X_\alpha, \alpha \in S.

To more effectively work with product spaces, we consider a couple of equivalent bases that we can define for the product topology. Let \mathcal{B}_\alpha be a base for the space X_\alpha. Consider B=\prod \limits_{\alpha \in S} B_\alpha such that there is a finite set F \subset S where B_\alpha \in \mathcal{B}_\alpha for each \alpha \in F and B_\alpha=X_\alpha for all \alpha \in S-F. The set of all such sets B is an equivalent base for the product topology.

Another equivalent base is defined using the projection maps. For each \alpha \in S, consider the map \pi_\alpha:\prod \limits_{\beta \in S} X_\beta \mapsto X_\alpha such that \pi_\alpha(f)=f_\alpha for each f in the product. In words, the function \pi_\alpha maps each point in the product space to its \alpha^{th} coordinate. The mapping \pi_\alpha is called the \alpha^{th} projection map. For each set U \subset X_\alpha, \pi_\alpha^{-1}(U) is the following set:

\pi_\alpha^{-1}(U)=\left\{f \in \prod \limits_{\beta \in S} X_\beta: \pi_\alpha(f)=f_\alpha \in U\right\}

Consider sets of the form \bigcap \limits_{\alpha \in F} \pi_\alpha^{-1}(U_\alpha) where F \subset S is finite and U_\alpha is open in X_\alpha for each \alpha \in F. The set of all such sets is another equivalent base for the product topology. If we only require that each U_\alpha \in \mathcal{B}_\alpha, a predetermined base for the coordinate space X_\alpha, we also obtain an equivalent base for the product topology.

Countable Product of First Countable Spaces
For i=1,2,3,\cdots, let X_i be a first countable space. We show that X=\prod \limits_{i=1}^{\infty}X_i is a first countable space.

Let \mathbb{N} be the set of positive integers. For each n \in \mathbb{N}, let [n]=\left\{1,2,\cdots,n\right\} and let \mathbb{N}^{[n]} be the set of all functions t:[n] \mapsto \mathbb{N}. For each i and each x \in X_i, let \mathcal{B}_x(i)=\left\{B_x(i,j): j \in \mathbb{N}\right\} be a countable local base at x.

Let f \in X=\prod \limits_{i=1}^{\infty}X_i. We wish to define a countable local base at f. For each n \in \mathbb{N}, define W_n to be:

W_n=\left\{\prod \limits_{i=1}^n B_{f(i)}(i,t(i)):t \in \mathbb{N}^{[n]}\right\}

Let W be the set of all subsets of the product space X of the following form:

\prod \limits_{j=1}^{\infty}V_j where there is some n \in \mathbb{N} such that \prod \limits_{j=1}^{n}V_j \in W_n and for all j>n, V_j=X_j.

Each W_n is countable and W is essentially the union of all the W_n. Thus W is countable. We claim that W is a local base at f. Let O \subset X be an open set containing f. We can assume that O=\prod \limits_{i=1}^\infty O_i where there is some n \in \mathbb{N} such that for each i \le n, O_i is open in X_i and for i>n, O_i=X_i.

For each i \le n, f(i) \in O_i. Choose some B_{f(i)}(i,t(i)) such that f(i) \in B_{f(i)}(i,t(i)) \subset O_i. Let V=\prod \limits_{i=1}^{\infty} V_i such that \prod \limits_{i=1}^{n} V_i=\prod \limits_{i=1}^n B_{f(i)}(i,t(i)) and V_i=X_i for all i>n. Then V \in W and f \in V \subset O. This completes the proof that X=\prod \limits_{i=1}^{\infty}X_i is a first countable space.

Uncountable Product
Let S be an uncountable index set. For \alpha \in S, let X_\alpha. We want to avoid the situation that all but countably many X_\alpha are one-point space. So we assume each coordinate space X_\alpha has at least two points, say, p_\alpha and q_\alpha with p_\alpha \ne q_\alpha. We show that X=\prod \limits_{\alpha \in S}X_\alpha is not first countable.

Let f \in \prod \limits_{\alpha \in S}X_\alpha. Let U_1,U_2, \cdots be open subsets of the product space such that for each i, f \in U_i. We show that there is some open set O such that f \in O and each U_i \nsubseteq O. For each i, there is a basic open set B_i=\bigcap \limits_{\alpha \in F_i} \pi_\alpha^{-1}(U_{\alpha,i}) such that f \in B_i \subset U_i.

Let F=F_1 \cup F_2 \cup \cdots. Since S is uncountable and F is countable, choose \gamma \in S-F. Since X_\gamma has at least two points p_\gamma and q_\gamma, choose one of them that is different from f_\gamma, say, p_\gamma. Choose two disjoint open subsets M_1 and M_2 of X_\gamma such that f_\gamma \in M_1 and p_\gamma \in M_2 of X_\gamma. Let O=\prod \limits_{\alpha \in S}O_\alpha such that O_\gamma=M_1 and O_\alpha=X_\alpha for all \alpha \ne \gamma. We have f \in O. For each i, there is g_i \in B_i \subset U_i such that g_i(\gamma)=p_\gamma. Thus each g_i \notin O. Thus there is no countable local base at f. Thus any product space with uncountably many factors, each of which has at least two points, is never first countable.

Reference

  1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

A note on metrization theorems for compact spaces

In a previous post (Metrization Theorems for Compact Spaces), three classic metrization theorems for compact spaces are discussed. The three theorems are: any Hausdorff compact space X is metrizable if any of the following holds:

  1. X has a countable network,
  2. X has a G_\delta diagonal,
  3. X has a point countable base.

The metrization results for conditions 2 and 3 hold for countably compact spaces as well. See the following posts:

Countably Compact Spaces with G-delta Diagonals
Metrization Theorems for Compact Spaces

In this post, we discuss another metrization theorem for compact spaces. We show that a compact Hausdorff space X is metrizable if and only if the function space C_p(X) is separable.

Let’s discuss the function space. Let Y by any Tychonoff space and let \mathbb{R} be the set of all real numbers. Let C(Y,\mathbb{R}) be the set of all real-valued continuous functions defined on Y. For any A \subset Y and for any V \subset \mathbb{R}, define [A,V]=\lbrace{f \in C(Y,\mathbb{R}): f(A) \subset V}\rbrace. If we restrict A to \lbrace{x}\rbrace and restrict V to open sets, then the set of all [A,V] is a subbase for a topology on C(Y,\mathbb{R}). This topology is called the pointwise convergence topology. The function space C(Y,\mathbb{R}) with this topology is denoted by C_p(Y).

It is a theorem that C_p(Y) is separable if and only if Y has a weaker topology that forms a separable metric space. The result on compact spaces is a corollary of this theorem.

Theorem. Let Y be a Tychonoff space with \tau being the topology. The following conditions are equivalent:

  1. C_p(Y) is separable.
  2. There is a topology \tau_1 \subset \tau such that (Y, \tau_1) is a separable metric space. 

Proof
1 \Rightarrow 2. Let D \subset C_p(Y) be a countable dense subspace. Let \mathcal{V} be the class of all bounded open intervals of \mathbb{R} with rational endpoints. Consider \mathcal{B}=\lbrace{f^{-1}(V): f \in D, V \in \mathcal{V}}\rbrace. Note that \mathcal{B} is a subbase for a topology \tau_1 on Y. Since \mathcal{B} is countable, the topology \tau_1 has a countable base and is thus separable and metrizable.

2 \Rightarrow 1. Let \tau_1 \subset \tau be a topology. Suppose that \tau_1 is generated by a countable base \mathcal{U}. As in 1 \Rightarrow 2, let \mathcal{V} be the class of all bounded open intervals of \mathbb{R} with rational endpoints. Let \mathcal{N} be the class of all finite intersections of the sets in the following collection of sets.

\displaystyle \lbrace{[A,V]: A \in \mathcal{U},V \in \mathcal{V}}\rbrace

Note that \mathcal{N} is countable. For each W \in \mathcal{N}, choose f_W \in W. We claim that \lbrace{f_W: W \in \mathcal{N}}\rbrace is a countable dense set of C_p(Y). To see this, let T=\bigcap_{i \le n} [A_i,V_i] be a basic open set in C_p(Y) where A_i=\lbrace{x_i}\rbrace. Fix f \in T. For each i, choose O_i \in \mathcal{U} such that x_i \in O_i and f(O_i) \subset V_i. Then W=\bigcap_{i \le n} [O_i,V_i] \in \mathcal{N}. Now, we have f_W \in T.

Corollary. Let X be a compact Hausdorff space. Then the following conditions are equivalent:

  1. X is metrizable.
  2. C_p(X) is separable.

Proof.
1 \Rightarrow 2. This follows from 2 \Rightarrow 1 in the above theorem.

2 \Rightarrow 1. This follows from 1 \Rightarrow 2 in the above theorem. Note that any compact Hausdorff space cannot have a strictly weaker (or coarser) Hausdorff topology. Thus if a compact Hausdorff space has a weaker metrizable topology, it must be metrizable.

The Evaluation Map

The evaluation map is a useful tool for embedding a space X into a product space. In this post we demonstrate that any Tychonoff space X can be embedded into a cube I^{\mathcal{K}} where I is the unit interval [0,1] and \mathcal{K} is some cardinal. Any regular space with a countable base (second-countable space) can also be embedded into the Hilbert cube I^{\omega} (Urysohn’s metrization theorem). The evaluation map also plays an important role in the theory of Cech-Stone compactification.

The Evaluation Map
Let X be a space. Let \displaystyle Y=\Pi_{\alpha \in A}Y_\alpha be a product space. For each y \in Y, we use the notation y=\langle y_\alpha \rangle_{\alpha \in A} to denote a point in the product space Y. Suppose we have a family of continuous functions \mathcal{F}=\lbrace{f_\alpha:\alpha \in A}\rbrace where f_\alpha:X \rightarrow Y_\alpha for each \alpha. Define a mapping E_{\mathcal{F}}:X \longrightarrow \Pi_{\alpha \in A}Y_\alpha as follows:

    For each x \in X, E_{\mathcal{F}}(x) is the point \langle f_\alpha(x) \rangle_{\alpha \in A} \in Y.

This mapping is called the evaluation map of the family of continuous functions \mathcal{F}=\lbrace{f_\alpha:\alpha \in A}\rbrace. If the family \mathcal{F} is understood, we may skip the subscript and use E to denote the evaluation map.

The family of continuous functions \mathcal{F} is said to separate points if for any two distinct points x,y \in X, there is a function f \in \mathcal{F} such that f(x) \neq f(y). The family of continuous functions \mathcal{F} is said to separate points from closed sets if for each point x \in X and for each closed set C \subset X with x \notin C, there is a function f \in \mathcal{F} such that f(x) \notin \overline{f(C)}.

Theorem 1. Given an evaluation map E_{\mathcal{F}}:X \longrightarrow \Pi_{\alpha \in A}Y_\alpha as defined above, the following conditions hold.

  1. The mapping E_{\mathcal{F}} is continuous.
  2. If the family of continuous functions \mathcal{F}=\lbrace{f_\alpha:\alpha \in A}\rbrace separates points, then E_{\mathcal{F}} is a one-to-one map.
  3. If the family of continuous functions \mathcal{F}=\lbrace{f_\alpha:\alpha \in A}\rbrace separates points from closed sets, then E_{\mathcal{F}} is a homeomorphism from X into the product space \displaystyle Y=\Pi_{\alpha \in A}Y_\alpha.

In this post, basic open sets in the product space \displaystyle Y=\Pi_{\alpha \in A}Y_\alpha are of the form \bigcap_{\alpha \in W} [\alpha,V_\alpha] where W \subset A is finite, for each \alpha \in W, V_\alpha is an open set in Y_\alpha and [\alpha,V_\alpha]=\lbrace{y \in Y:y_\alpha \in V_\alpha}\rbrace.

Proof of 1. We show that E_{\mathcal{F}} is continuous at each x \in X. Let x \in X. Let h=\langle f_\alpha(x) \rangle_{\alpha \in A} and let h \in V \cap E_{\mathcal{F}}(X) where V=\bigcap_{\alpha \in W} [\alpha,V_\alpha] is a basic open set. Consider U=\bigcap_{\alpha \in W} f_\alpha^{-1}(V_\alpha). It is easy to verify that x \in U and E_{\mathcal{F}}(U) \subset V\cap E_{\mathcal{F}}(X).

Proof of 2. Let x,y \in X be distinct points. There is \alpha \in A  such that f_\alpha(x) \neq f_\alpha(y). Clearly, E_{\mathcal{F}}(x)= \langle f_\beta(x) \rangle_{\beta \in A} \neq E_{\mathcal{F}}(y)=\langle f_\beta(y) \rangle_{\beta \in A}.

Proof of 3. Note that by condition 2 in this theorem, the map E_{\mathcal{F}} is one-to-one. It suffices to show that E_{\mathcal{F}} is an open map. Let U \subset X be open. We show that E_{\mathcal{F}}(U) is open in E_{\mathcal{F}}(X). To this end, let \langle f_\alpha(x) \rangle_{\alpha \in A} \in E_{\mathcal{F}}(U). Then x \in U. Since \mathcal{F} separates points from closed sets, there is some \beta such that f_\beta(x) \notin \overline{f_\beta(X-U)}. Let V_\beta=Y_\beta-\overline{f_\beta(X-U)}. Then \langle f_\alpha(x) \rangle_{\alpha \in A} \in [\beta,V_\beta] \cap E_{\mathcal{F}}(X)=W_\beta. We show that W_\beta \subset E_{\mathcal{F}}(U). For each \langle f_\alpha(y) \rangle_{\alpha \in A} \in W_\beta, we have f_\beta(y) \notin \overline{f_\beta(X-U)}. If y \notin U, then f_\beta(y) \in f_\beta(X-U), a contradiction. So we have y \in U and this means that \langle f_\alpha(y) \rangle_{\alpha \in A} \in E_{\mathcal{F}}(U). It follows that W_\beta \subset E_{\mathcal{F}}(U).

Some Applications

A space X is a Tychonoff space (also known as completely regular space) if for each x \in X and for each closed set C \subset X where x \notin C, there is a continuous function f:X \rightarrow I such that f(x)=1 and f(y)=0 for all y \in C. The following is a corollary to theorem 1.

Corollary 1. Any Tychonoff space can be embedded in a cube I^{\mathcal{K}}.

Proof. Let \mathcal{F} be the family of all continuous functions from the Tychonoff space X into the unit interval I. By the definition of Tychonoff space, \mathcal{F} separates points from closed sets. By theorem 1, the evaluation map E_{\mathcal{F}} is a homeomorphism from X into the cube I^{\mathcal{K}} where \mathcal{K}=\lvert \mathcal{F} \lvert.

We now turn our attention to regular second countable space. Having a countable base has many strong properties, one of which is that it can be embedded into the Hilbert Cube I^{\omega}=I^{\aleph_0}. Before we prove this, observe that any regular space with a countable base is a regular Lindelof space. Furthermore, the property of having a countable base is hereditary. Thus a regular space with a countable base is hereditarily Lindelof (hence perfectly normal). The Vendenisoff Theorem states that in a perfectly normal space, every closed set is a zero-set (i.e. every open set is a cozero-set). So we make use of this theorem to obtain continuous functions that separate points from closed sets. There is a proof of The Vendenisoff Theorem in this blog. A set Z \subset X is a zero-set in the space X if there is a continuous function f:X \rightarrow I such that f^{-1}(0)=Z. A set W \subset X is a cozero-set if X-W is a zero-set. We are now ready to prove one part of the Urysohn’s metrization theorem.

Urysohn’s metrization theorem. The following conditions are equivalent.

  1. The space X is a regular space with a countable base.
  2. The space X can be embedded into the Hilbert cube I^{\aleph_0}.
  3. The space X is a separable metric space.

We prove the direction 1 \Rightarrow 2. Let \lbrace{B_0,B_1,B_2,...}\rbrace be a countable base for the regular space X. Based on the preceding discussion, X is perfectly normal. By the Vendenisoff Theorem, for each nX-B_n is a zero-set. Thus for each n, there is a continuous function f_n:X \rightarrow I such that f_n^{-1}(0)=X-B_n and f_n^{-1}((0,1])=B_n. Let \mathcal{F}=\lbrace{f_0,f_1,f_2,...}\rbrace. It is easy to verify that \mathcal{F} separates points from closed sets. Thus the evaluation map E_{\mathcal{F}} is a homeomorphism from X into I^{\aleph_0}.