# The Michael Line and the Continuum Hypothesis

There exist a Lindelof space and a separable metric space such that their Cartesian product is not normal (discussed in the post “Bernstein Sets and the Michael Line”). The separable metric space is a Bernstein set, a subspace of the real line that is far from being a complete metric space. However, this example is constructed without using any additional set theory axiom beyond the Zermelo-Fraenkel axioms plus the axiom of choice (abbreviated ZFC). A natural question is whether there exists a Lindelof space and a complete metric space such that their product is not normal. In particular, does there exist a Lindelof space $L$ such that the product of $L$ with the space of all irrational numbers is not normal? As of the writing of this post, it is still unknown that such a Lindelof space can exist in just ZFC alone without applying additional set theory axiom. However, such a Lindelof space can be constructed from various additional axioms (e.g. continuum hypothesis or Martin’s axiom). In this post, we present an example of such construction using the continuum hypothesis (the statement that the cardinality of the real line is the same as the first uncountable cardinal $\aleph_1$).

Let $\mathbb{M}$ be the Michael line. Let $\mathbb{P}$ be the set of irrational numbers with the usual topology inherited from the real line. It is a classical result that the product $\mathbb{M} \times \mathbb{P}$ is not normal (see “Michael Line Basics”). The Lindelof example we wish to discuss is an uncountable Lindelof subspace $L$ of $\mathbb{M}$ such that $L$ contains the set $\mathbb{Q}$ of rational numbers. The same proof that $\mathbb{M} \times \mathbb{P}$ is not normal will show that $L \times \mathbb{P}$ is not normal.

See the following posts for a basic discussion of the Michael line:

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Luzin Sets

The Lindelof space $X$ we want to find is a subset of the real line that is called a Luzin set. Before defining Luzin sets, recall some definitions. Let $Y$ be a space. Let $A \subset Y$. The set $A$ is said to be nowhere dense in $Y$ if for every non-empty open subset $U$ of $Y$, there is a non-empty open subset $V$ of $Y$ such that $V \subset U$ and $V$ misses $A$ (equivalently, the closure of $A$ has no interior). The set $A$ is of first category in $Y$ if it is the union of countably many nowhere dense sets.

To define Luzin sets, we focus on the Euclidean space $\mathbb{R}$. Let $A \subset \mathbb{R}$. The set $A$ is said to be a Luzin set if for every set $W \subset \mathbb{R}$ that is of first category in the real line, $A \cap W$ is at most countable. The Russian mathematician Luzin in 1914 constructed such an uncountable Luzin set using continuum hypothesis (CH). A good reference for Luzin sets is [4]. We have the following theorem.

Theorem 1
Assume CH. There exists an uncountable Luzin set.

Proof of Theorem 1
There are continuum many closed nowhere dense subsets of the real line. Since we assume the continuum hypothesis, we can enumerate these sets in a sequence of length $\omega_1$. Let $\left\{F_\alpha: \alpha < \omega_1 \right\}$ be the set of all closed nowhere dense sets in the real line. Choose a real number $x_0 \notin F_0$ to start. For each $\alpha$ with $0 < \alpha <\omega_1$, choose a real number $x_\alpha$ not in the following set:

$\left\{x_\beta: \beta<\alpha \right\} \cup \bigcup \limits_{\beta<\alpha} F_\beta$

The above set is a countable union of closed nowhere dense sets of the real line. As a complete metric space, the real line cannot be of first category. In fact, according to the Baire category theorem, the complement of a set of first category (such as the one described above) is dense in the real line. So such an $x_\alpha$ can always be selected at each $\alpha<\omega_1$. Then $X=\left\{x_\alpha: \alpha<\omega_1 \right\}$ is a Luzin set. $\blacksquare$

Now that we have a way of constructing an uncountable Luzin sets, the following observations provide some useful facts for our problem at hand.

Nowhere dense sets and sets of first category are "thin" sets. Any "thin" set can intersect with a Luzin set with only countably many points. Thus any "co-thin" set contains all but countably many points of a Luzin set. For example, let $A$ be an uncountable Luzin set. Then if $F$ is a closed nowhere dense set in the real line, then $\mathbb{R}-F$ contains all but countably many points of $A$. Furthermore, if $F_1,F_2,F_3,\cdots,$ are closed nowhere dense subsets of the real line, then $\mathbb{R}- \bigcup \limits_{i=1}^\infty F_i$ contains all but countably many points of the Luzin set $A$.

Note that the set $\mathbb{R}-F$ in the preceding paragraph is a dense open set. Thus the complement of a closed nowhere dense set is a dense open set. Note that the set $\mathbb{R}- \bigcup \limits_{i=1}^\infty F_i$ in the preceding paragraph is a dense $G_\delta$-set. Thus the complement of the union of countably many closed nowhere dense sets is a dense $G_\delta$-set. Thus the observation in the preceding paragraph gives the following proposition:

Proposition 2
Given an uncountable Luzin set $A$ and given a dense $G_\delta$ subset $H$ of the real line, $H$ contains all but countably many points of $A$.

In fact, Proposition 2 not only hold in the real line, it also holds in any uncountable dense subset of the real line.

Proposition 3
Let $A$ be an uncountable Luzin set. Let $Y \subset \mathbb{R}$ be uncountable and dense in the real line such that $A \cap Y$ is uncountable. Given a dense $G_\delta$ subset $H$ of $Y$, $H$ contains all but countably many points of $A \cap Y$.

Proof of Proposition 3
We want to show that $Y-H$ can only contain countably many points of $A$. Let $H=\bigcap \limits_{i=1}^\infty O_i$ where each $O_i$ is open and dense in $Y$. Then for each $i$, let $U_i$ be open in the real line such that $U_i \cap Y=O_i$. Each $U_i$ is open and dense in the real line. Thus $H^*=\bigcap \limits_{i=1}^\infty U_i$ contains all but countably many points of the Luzin set $A$. Note the following set inclusion:

$H=\bigcap \limits_{i=1}^\infty U_i \cap Y=\bigcap \limits_{i=1}^\infty O_i \subset \bigcap \limits_{i=1}^\infty U_i=H^*$

Suppose that $Y-H$ contains uncountably many points of $A$. Then these points, except for countably many points, must belong to $H^*=\bigcap \limits_{i=1}^\infty U_i$. The above set inclusion shows that these points must belong to $H$ too, a contradiction. Thus $Y-H$ can only contain countably many points of $A$, equivalently the $G_\delta$-set $H$ contains all but countably many points of $A \cap Y$. $\blacksquare$

The following proposition follows from Proposition 3 and is a useful fact that will help us see that the product of an uncountable Luzin set and $\mathbb{P}$ is not normal.

Proposition 4
Let $Y$ be an uncountable Luzin set such that $\mathbb{Q} \subset Y$. Then $Y-\mathbb{Q}$ cannot be an $F_\sigma$-set in the Euclidean space $Y$, equivalently $\mathbb{Q}$ cannot be a $G_\delta$-set in the space $Y$.

Proof of Proposition 4
By Proposition 3, any dense $G_\delta$-subset of $Y$ must be co-countable. $\blacksquare$

The following proposition is another useful observation about Luzin sets. Let $A \subset \mathbb{R}$. Let $D \subset \mathbb{R}$ be a countable dense subset of the real line. The set $A$ is said to be concentrated about $D$ if for every open subset $O$ of the real line such that $D \subset O$, $O$ contains all but countably many points of $A$. The following proposition can be readily checked based on the definition of Luzin sets.

Proposition 5
For any $A \subset \mathbb{R}$, $A$ is a Luzin set if and only if $A$ is concentrated about every countable dense subset of the real line.

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Lindelof Subspace of The Michael Line

Let $A$ be an uncountable Luzin set. We can assume that $A$ is dense in the real line. If not, just add a countble subset of $\mathbb{P}$ that is dense in the real line. Let $L=A \cup \mathbb{Q}$. It is clear that adding countably many points to a Luzin set still results in a Luzin set. Thus $L$ is also a Luzin set. Now consider $L$ as a subspace of the Michael line $\mathbb{M}$. Then points of $L-\mathbb{Q}$ are discrete and points in $\mathbb{Q}$ have Euclidean open neighborhoods. By Proposition 5, the set $L$ is concentrated about every countable dense subset of the real line. In particular, it is concentrated about $\mathbb{Q}$. Thus as a subspace of the Michael line, $L$ is a Lindelof space, since every open set containing $\mathbb{Q}$ contains all but countably many points of $L$.

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The Non-Normal Product $L \times \mathbb{P}$

We highlight the following two facts about the Luzin set $L=A \cup \mathbb{Q}$ as discussed in the preceding section.

• $L-\mathbb{Q}$ is not an $F_\sigma$-set in $L$ (as Euclidean space).
• $A=L-\mathbb{Q}$ is dense in the real line.

The first bullet point follows from Proposition 4. The second bullet point is clear since we assume the Luzin set $A$ we start with is dense. Recall that when thinking of $L$ as a subspace of the Michael line, $L-\mathbb{Q}$ are isolated and $\mathbb{Q}$ retains the usual real line open sets. Because of the above two bullet points, $L \times \mathbb{P}$ is not normal. The proof that $L \times \mathbb{P}$ is not normal is the corollary of the proof that $\mathbb{M} \times \mathbb{P}$ is not normal. Note that in the proof for showing $\mathbb{M} \times \mathbb{P}$ is not normal, the two crucial points about the proof are that the isolated points of the Michael line cannot be an $F_\sigma$-set and are dense in the real line (found in “Michael Line Basics”).

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Michael Space

The example $L \times \mathbb{P}$ that we construct here was hinted in footnote 4 in [6]. In a later publication, E. Michael constructed an uncountable Lindelof subspace of the Michael line (see Lemma 3.1 in [5]). That construction should produce a similar set as the Luzin sets since the approach in [5] is a mirror image of the Luzin set construction. The approach in the Luzin set construction in Theorem 1 is to pick points not in the union of countably many closed nowhere dense sets, while the approach in [5] was to pick points in dense $G_\delta$-sets in a transfinite induction process.

A Michael space is a Lindelof space whose product with $\mathbb{P}$ is not normal. The example shown here shows that under CH, there exists a Michael space. However, the question of whether there exists a Michael space in ZFC is still unsolved. This is called the Michael problem. A recent mention of this unsolved problem is [3] (page 160). A Michael space can also be constructed using Martin’s axiom (see [1]).

A space is said to be a productively Lindelof space if its product with every Lindelof space is Lindelof. Is $\mathbb{P}$ a productively Lindelof space? As we see here, under CH the answer is no. Another way of looking at the Michael problem: is it possible to show that $\mathbb{P}$ is not productively Lindelof in ZFC alone?

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Reference

1. Alster, K., The product of a Lindelof space with the space of irrationals under Martin’s Axiom, Proc. Amer. Math. Soc., 110 (1990) 543-547.
2. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
3. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
4. Miller, A. W., Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 201-233, 1984.
5. Michael, E., Paracompactness and the Lindelof property in Finite and Countable Cartesian Products, Compositio Math. 23 (1971) 199-214.
6. Michael, E., The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc., 69 (1963) 375-376.
7. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

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$\copyright \ \ 2012$

# Bernstein Sets and the Michael Line

Let $\mathbb{M}$ be the Michael line and let $\mathbb{P}$ be the set of all irrational numbers with the Euclidean topology. In the post called “Michael Line Basics”, we show that the product $\mathbb{M} \times \mathbb{P}$ is not normal. This is a classic counterexample showing that the product of two paracompact spaces need not be normal even when one of the factors is a complete metric space. The Michael line $\mathbb{M}$ is not Lindelof. A natural question is: can the first factor be made a Lindelof space? In this post, as an application of Bernstein sets, we present a non-normal product space where one factor is Lindelof and the other factor is a separable metric space. It is interesting to note that while one factor is upgraded (from paracompact to Lindelof), the other factor is downgraded (from a complete metric space to just a separable metric space).

Bernstein sets have been discussed previously in this blog. They are special subsets of the real line and with the Euclidean subspace topology, they are spaces in which the Banach-Mazur game is undecidable (see the post “Bernstein Sets Are Baire Spaces”). A Bernstein set is a subset $B$ of the real line such that every uncountable closed subset of the real line has non-empty intersection with both $B$ and the complement of $B$.

Bernstein sets are constructed by transfinite induction. The procedure starts by ordering all uncountable closed subsets of the real line in a sequence of length that is as long as the cardinality of continuum. To see how Bernstein sets are constructed, see the post “Bernstein Sets Are Baire Spaces”.

After we discuss a generalization of the definition of the Michael line, we discuss the non-normal product space based on Bernstein sets.
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Generalizing the Michael Line

Let $\mathbb{R}$ be the real number line. Let $\mathbb{P}$ be the set of all irrational numbers and let $\mathbb{Q}=\mathbb{R}-\mathbb{P}$. Recall that the Michael line is the real line $\mathbb{R}$ topologized by letting points in $\mathbb{P}$ discrete and letting points in $\mathbb{Q}$ retain their usual open neighborhoods. We can carry out the same process on any partition of the real number line.

Let $D$ and $E$ be disjoint sets such that $\mathbb{R}=D \cup E$ where the set $E$ is dense in the real line. The intention is to make $D$ the discrete part and $E$ the Euclidean part. In other words, we topologize $\mathbb{R}$ be letting points in $D$ discrete and letting points in $E$ retain their Euclidean open sets. Let $X_D$ denote the resulting topological space. For the lack of a better term, we call the space $X_D$ the modified Michael line. An open set in the space $X_D$ is of the form $U \cup V$ where $U$ is a Euclidean open subset of the real line and $V \subset D$. We have the following result:

Proposition
Suppose that $D$ is not an $F_\sigma$-set in the Euclidean real line and that $D$ is dense in the Euclidean real line. Then the product space $X_D \times D$ is not normal (the second factor $D$ is considered a subspace of the Euclidean real line).

In the post “Michael Line Basics”, we give a proof that $\mathbb{M} \times \mathbb{P}$ is not normal. This proof hinges on the same two facts about the set $D$ in the hypothesis in the above proposition. Thus the proof for the above proposition is just like the one for $\mathbb{M} \times \mathbb{P}$. Whenever we topologize the modified Michael line by using a non-$F_\sigma$-set as the discrete part, we can always be certain that we have a non-normal product as indicated here.

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Non-Normal Product Space

Let $B$ be any Bernstein set. The set $B$ is clearly not an $F_\sigma$-set in the real line and is clearly dense in the real line. Then $X_B \times B$ is not normal. Note that in $X_B$, the set $B$ is discrete and its complement $\mathbb{R}-B$ has the usual topology. To see that $X_B$ is Lindelof, note that any open cover of $X_B$ has a countable subcollection that covers $\mathbb{R}-B$. This countable subcollection consists of Euclidean open sets. Furthermore, the complement of the union of these countably many Euclidean open sets must contain all but countably many points of the Bernstein set $B$ (otherwise there would be an uncountable Euclidean closed set that misses $B$).

As commented at the beginning, in obtaining this non-normal product space, one factor is enhanced at the expense of the other factor (one is made Lindelof while the other is no longer a complete metric space). Even though any Bernstein set (with the Euclidean topology) is a separable metric space, it cannot be completely metrizable. Any completely metrizable subset of the real line must be a $G_\delta$-set in the real line. Furthermore any uncountable $G_\delta$ subset of the real line must contain a Cantor set and thus cannot be a Bernstein set.

A similar example to $X_B \times B$ is presented in E. Michael’s paper (see [3]). It is hinted in footnote 4 of that paper that with the additional assumption of continuum hypothesis (CH), one can have a non-normal product space where one factor is a Lindelof space and the second factor is the space of irrationals. So with an additional set-theoretic assumption, we can keep one factor from losing complete metrizability. For this construction, see point (d) in Example 3.2 of [2].

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A Brief Remark

Note that the Lindelof space $X_B$ presented here is not hereditarily Lindelof, since it has uncountably many isolated points. Can a hereditarily Lindelof example be constructed such that its product with a particular separable metric space is not normal? The answer is no. The product of a hereditarily Lindelof space and any separable metric space is hereditarily Lindelof (see Result 4 in the post Cartesian Products of Two Paracompact Spaces – Continued).

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Reference

1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
2. Michael, E., Paracompactness and the Lindelof property in Finite and Countable Cartesian Products, Compositio Math. 23 (1971) 199-214.
3. Michael, E., The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc., 69 (1963) 375-376.
4. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

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$\copyright \ \ 2012$

# Michael Line Basics

Like the Sorgenfrey line, the Michael line is a classic counterexample that is covered in standard topology textbooks and in first year topology courses. This easily accessible example helps transition students from the familiar setting of the Euclidean topology on the real line to more abstract topological spaces. One of the most famous results regarding the Michael line is that the product of the Michael line with the space of the irrational numbers is not normal. Thus it is an important example in demonstrating the pathology in products of paracompact spaces. The product of two paracompact spaces does not even have be to be normal, even when one of the factors is a complete metric space. In this post, we discuss this classical result and various other basic results of the Michael line.

Let $\mathbb{R}$ be the real number line. Let $\mathbb{P}$ be the set of all irrational numbers. Let $\mathbb{Q}=\mathbb{R}-\mathbb{P}$, the set of all rational numbers. Let $\tau$ be the usual topology of the real line $\mathbb{R}$. The following is a base that defines a topology on $\mathbb{R}$.

$\mathcal{B}=\tau \cup \left\{\left\{ x \right\}: x \in \mathbb{P}\right\}$

The real line with the topology generated by $\mathcal{B}$ is called the Michael line and is denoted by $\mathbb{M}$. In essense, in $\mathbb{M}$, points in $\mathbb{P}$ are made isolated and points in $\mathbb{Q}$ retain the usual Euclidean open sets.

The Euclidean topology $\tau$ is coarser (weaker) than the Michael line topology (i.e. $\tau$ being a subset of the Michael line topology). Thus the Michael line is Hausdorff. Since the Michael line topology contains a metrizable topology, $\mathbb{M}$ is submetrizable (submetrized by the Euclidean topology). It is clear that $\mathbb{M}$ is first countable. Having uncountably many isolated points, the Michael line does not have the countable chain condition (thus is not separable). The following points are discussed in more details.

1. The space $\mathbb{M}$ is paracompact.
2. The space $\mathbb{M}$ is not Lindelof.
3. The extent of the space $\mathbb{M}$ is $c$ where $c$ is the cardinality of the real line.
4. The space $\mathbb{M}$ is not locally compact.
5. The space $\mathbb{M}$ is not perfectly normal, thus not metrizable.
6. The space $\mathbb{M}$ is not a Moore space, but has a $G_\delta$-diagonal.
7. The product $\mathbb{M} \times \mathbb{P}$ is not normal where $\mathbb{P}$ has the usual topology.
8. The product $\mathbb{M} \times \mathbb{P}$ is metacompact.
9. The space $\mathbb{M}$ has a point-countable base.
10. For each $n=1,2,3,\cdots$, the product $\mathbb{M}^n$ is paracompact.
11. The product $\mathbb{M}^\omega$ is not normal.
12. There exist a Lindelof space $L$ and a separable metric space $W$ such that $L \times W$ is not normal.

Results 10, 11 and 12 are shown in some subsequent posts.

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Baire Category Theorem

Before discussing the Michael line in greater details, we point out one connection between the Michael line topology and the Euclidean topology on the real line. The Michael line topology on $\mathbb{Q}$ coincides with the Euclidean topology on $\mathbb{Q}$. A set is said to be a $G_\delta$-set if it is the intersection of countably many open sets. By the Baire category theorem, the set $\mathbb{Q}$ is not a $G_\delta$-set in the Euclidean real line (see the section called “Discussion of the Above Question” in the post A Question About The Rational Numbers). Thus the set $\mathbb{Q}$ is not a $G_\delta$-set in the Michael line. This fact is used in Result 5.

The fact that $\mathbb{Q}$ is not a $G_\delta$-set in the Euclidean real line implies that $\mathbb{P}$ is not an $F_\sigma$-set in the Euclidean real line. This fact is used in Result 7.

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Result 1

Let $\mathcal{U}$ be an open cover of $\mathbb{M}$. We proceed to derive a locally finite open refinement $\mathcal{V}$ of $\mathcal{U}$. Recall that $\tau$ is the usual topology on $\mathbb{R}$. Assume that $\mathcal{U}$ consists of open sets in the base $\mathcal{B}$. Let $\mathcal{U}_\tau=\mathcal{U} \cap \tau$. Let $Y=\cup \mathcal{U}_\tau$. Note that $Y$ is a Euclidean open subspace of the real line (hence it is paracompact). Then there is $\mathcal{V}_\tau \subset \tau$ such that $\mathcal{V}_\tau$ is a locally finite open refinement $\mathcal{V}_\tau$ of $\mathcal{U}_\tau$ and such that $\mathcal{V}_\tau$ covers $Y$ (locally finite in the Euclidean sense). Then add to $\mathcal{V}_\tau$ all singleton sets $\left\{ x \right\}$ where $x \in \mathbb{M}-Y$ and let $\mathcal{V}$ denote the resulting open collection.

The resulting $\mathcal{V}$ is a locally finite open collection in the Michael line $\mathbb{M}$. Furthermore, $\mathcal{V}$ is also a refinement of the original open cover $\mathcal{U}$. $\blacksquare$

A similar argument shows that $\mathbb{M}$ is hereditarily paracompact.

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Result 2

To see that $\mathbb{M}$ is not Lindelof, observe that there exist Euclidean uncountable closed sets consisting entirely of irrational numbers (i.e. points in $\mathbb{P}$). For example, it is possible to construct a Cantor set entirely within $\mathbb{P}$.

Let $C$ be an uncountable Euclidean closed set consisting entirely of irrational numbers. Then this set $C$ is an uncountable closed and discrete set in $\mathbb{M}$. In any Lindelof space, there exists no uncountable closed and discrete subset. Thus the Michael line $\mathbb{M}$ cannot be Lindelof. $\blacksquare$

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Result 3

The argument in Result 2 indicates a more general result. First, a brief discussion of the cardinal function extent. The extent of a space $X$ is the smallest infinite cardinal number $\mathcal{K}$ such that every closed and discrete set in $X$ has cardinality $\le \mathcal{K}$. The extent of the space $X$ is denoted by $e(X)$. When the cardinal number $e(X)$ is $e(X)=\aleph_0$ (the first infinite cardinal number), the space $X$ is said to have countable extent, meaning that in this space any closed and discrete set must be countably infinite or finite. When $e(X)>\aleph_0$, there are uncountable closed and discrete subsets in the space.

It is straightforward to see that if a space $X$ is Lindelof, the extent is $e(X)=\aleph_0$. However, the converse is not true.

The argument in Result 2 exhibits a closed and discrete subset of $\mathbb{M}$ of cardinality $c$. Thus we have $e(\mathbb{M})=c$. $\blacksquare$

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Result 4

The Michael line $\mathbb{M}$ is not locally compact at all rational numbers. Observe that the Michael line closure of any Euclidean open interval is not compact in $\mathbb{M}$. $\blacksquare$

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Result 5

A set is said to be a $G_\delta$-set if it is the intersection of countably many open sets. A space is perfectly normal if it is a normal space with the additional property that every closed set is a $G_\delta$-set. In the Michael line $\mathbb{M}$, the set $\mathbb{Q}$ of rational numbers is a closed set. Yet, $\mathbb{Q}$ is not a $G_\delta$-set in the Michael line (see the discussion above on the Baire category theorem). Thus $\mathbb{M}$ is not perfectly normal and hence not a metrizable space. $\blacksquare$

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Result 6

The diagonal of a space $X$ is the subset of its square $X \times X$ that is defined by $\Delta=\left\{(x,x): x \in X \right\}$. If the space is Hausdorff, the diagonal is always a closed set in the square. If $\Delta$ is a $G_\delta$-set in $X \times X$, the space $X$ is said to have a $G_\delta$-diagonal. It is well known that any metric space has $G_\delta$-diagonal. Since $\mathbb{M}$ is submetrizable (submetrized by the usual topology of the real line), it has a $G_\delta$-diagonal too.

Any Moore space has a $G_\delta$-diagonal. However, the Michael line is an example of a space with $G_\delta$-diagonal but is not a Moore space. Paracompact Moore spaces are metrizable. Thus $\mathbb{M}$ is not a Moore space. For a more detailed discussion about Moore spaces, see Sorgenfrey Line is not a Moore Space. $\blacksquare$

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Result 7

We now show that $\mathbb{M} \times \mathbb{P}$ is not normal where $\mathbb{P}$ has the usual topology. In this proof, the following two facts are crucial:

• The set $\mathbb{P}$ is not an $F_\sigma$-set in the real line.
• The set $\mathbb{P}$ is dense in the real line.

Let $H$ and $K$ be defined by the following:

$H=\left\{(x,x): x \in \mathbb{P} \right\}$
$K=\mathbb{Q} \times \mathbb{P}$.

The sets $H$ and $K$ are disjoint closed sets in $\mathbb{M} \times \mathbb{P}$. We show that they cannot be separated by disjoint open sets. To this end, let $H \subset U$ and $K \subset V$ where $U$ and $V$ are open sets in $\mathbb{M} \times \mathbb{P}$.

To make the notation easier, for the remainder of the proof of Result 7, by an open interval $(a,b)$, we mean the set of all real numbers $t$ with $a. By $(a,b)^*$, we mean $(a,b) \cap \mathbb{P}$. For each $x \in \mathbb{P}$, choose an open interval $U_x=(a,b)^*$ such that $\left\{x \right\} \times U_x \subset U$. We also assume that $x$ is the midpoint of the open interval $U_x$. For each positive integer $k$, let $P_k$ be defined by:

$P_k=\left\{x \in \mathbb{P}: \text{ length of } U_x > \frac{1}{k} \right\}$

Note that $\mathbb{P}=\bigcup \limits_{k=1}^\infty P_k$. For each $k$, let $T_k=\overline{P_k}$ (Euclidean closure in the real line). It is clear that $\bigcup \limits_{k=1}^\infty P_k \subset \bigcup \limits_{k=1}^\infty T_k$. On the other hand, $\bigcup \limits_{k=1}^\infty T_k \not\subset \bigcup \limits_{k=1}^\infty P_k=\mathbb{P}$ (otherwise $\mathbb{P}$ would be an $F_\sigma$-set in the real line). So there exists $T_n=\overline{P_n}$ such that $\overline{P_n} \not\subset \mathbb{P}$. So choose a rational number $r$ such that $r \in \overline{P_n}$.

Choose a positive integer $j$ such that $\frac{2}{j}<\frac{1}{n}$. Since $\mathbb{P}$ is dense in the real line, choose $y \in \mathbb{P}$ such that $r-\frac{1}{j}. Now we have $(r,y) \in K \subset V$. Choose another integer $m$ such that $\frac{1}{m}<\frac{1}{j}$ and $(r-\frac{1}{m},r+\frac{1}{m}) \times (y-\frac{1}{m},y+\frac{1}{m})^* \subset V$.

Since $r \in \overline{P_n}$, choose $x \in \mathbb{P}$ such that $r-\frac{1}{m}. Now it is clear that $(x,y) \in V$. The following inequalities show that $(x,y) \in U$.

$\lvert x-y \lvert \le \lvert x-r \lvert + \lvert r-y \lvert < \frac{1}{m}+\frac{1}{j} \le \frac{2}{j} < \frac{1}{n}$

The open interval $U_x$ is chosen to have length $> \frac{1}{n}$. Since $\lvert x-y \lvert < \frac{1}{n}$, $y \in U_x$. Thus $(x,y) \in \left\{ x \right\} \times U_x \subset U$. We have shown that $U \cap V \ne \varnothing$. Thus $\mathbb{M} \times \mathbb{P}$ is not normal. $\blacksquare$

Remark
As indicated above, the proof of Result 7 hinges on two facts about $\mathbb{P}$, namely that it is not an $F_\sigma$-set in the real line and it is dense in the real line. We can modify the construction of the Michael line by using other partition of the real line (where one set is isolated and its complement retains the usual topology). As long as the set $D$ that is isolated is not an $F_\sigma$-set in the real line and is dense in the real line, the same proof will show that the product of the modified Michael line and the space $D$ (with the usual topology) is not normal. This will be how Result 12 is derived.

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Result 8

The product $\mathbb{M} \times \mathbb{P}$ is not paracompact since it is not normal. However, $\mathbb{M} \times \mathbb{P}$ is metacompact.

A collection of subsets of a space $X$ is said to be point-finite if every point of $X$ belongs to only finitely many sets in the collection. A space $X$ is said to be metacompact if each open cover of $X$ has an open refinement that is a point-finite collection.

Note that $\mathbb{M} \times \mathbb{P}=(\mathbb{P} \times \mathbb{P}) \cup (\mathbb{Q} \times \mathbb{P})$. The first $\mathbb{P}$ in $\mathbb{P} \times \mathbb{P}$ is discrete (a subspace of the Michael line) and the second $\mathbb{P}$ has the Euclidean topology.

Let $\mathcal{U}$ be an open cover of $\mathbb{M} \times \mathbb{P}$. For each $a=(x,y) \in \mathbb{Q} \times \mathbb{P}$, choose $U_a \in \mathcal{U}$ such that $a \in U_a$. We can assume that $U_a=A \times B$ where $A$ is a usual open interval in $\mathbb{R}$ and $B$ is a usual open interval in $\mathbb{P}$. Let $\mathcal{G}=\lbrace{U_a:a \in \mathbb{Q} \times \mathbb{P}}\rbrace$.

Fix $x \in \mathbb{P}$. For each $b=(x,y) \in \lbrace{x}\rbrace \times \mathbb{P}$, choose some $U_b \in \mathcal{U}$ such that $b \in U_b$. We can assume that $U_b=\lbrace{x}\rbrace \times B$ where $B$ is a usual open interval in $\mathbb{P}$. Let $\mathcal{H}_x=\lbrace{U_b:b \in \lbrace{x}\rbrace \times \mathbb{P}}\rbrace$.

As a subspace of the Euclidean plane, $\bigcup \mathcal{G}$ is metacompact. So there is a point-finite open refinement $\mathcal{W}$ of $\mathcal{G}$. For each $x \in \mathbb{P}$, $\mathcal{H}_x$ has a point-finite open refinement $\mathcal{I}_x$. Let $\mathcal{V}$ be the union of $\mathcal{W}$ and all the $\mathcal{I}_x$ where $x \in \mathbb{P}$. Then $\mathcal{V}$ is a point-finite open refinement of $\mathcal{U}$.

Note that the point-finite open refinement $\mathcal{V}$ may not be locally finite. The vertical open intervals in $\lbrace{x}\rbrace \times \mathbb{P}$, $x \in \mathbb{P}$ can “converge” to a point in $\mathbb{Q} \times \mathbb{P}$. Thus, metacompactness is the best we can hope for. $\blacksquare$

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Result 9

A collection of sets is said to be point-countable if every point in the space belongs to at most countably many sets in the collection. A base $\mathcal{G}$ for a space $X$ is said to be a point-countable base if $\mathcal{G}$, in addition to being a base for the space $X$, is also a point-countable collection of sets. The Michael line is an example of a space that has a point-countable base and that is not metrizable. The following is a point-countable base for $\mathbb{M}$:

$\mathcal{G}=\mathcal{H} \cup \left\{\left\{ x \right\}: x \in \mathbb{P}\right\}$

where $\mathcal{H}$ is the set of all Euclidean open intervals with rational endpoints. One reason for the interest in point-countable base is that any countable compact space (hence any compact space) with a point-countable base is metrizable (see Metrization Theorems for Compact Spaces).

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Reference

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

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$\copyright \ \ 2012$

# Stone-Cech Compactification of the Integers – Basic Facts

This is another post Stone-Cech compactification. The links for other posts on Stone-Cech compactification can be found below. In this post, we prove a few basic facts about $\beta \omega$, the Stone-Cech compactification of the discrete space of the non-negative integers, $\omega=\left\{0,1,2,3,\cdots \right\}$. We use several characterizations of Stone-Cech compactification to find out what $\beta \omega$ is like. These characterizations are proved in the blog posts listed below. Let $c$ denote the cardinality of the real line $\mathbb{R}$. We prove the following facts.

1. The cardinality of $\beta \omega$ is $2^c$.
2. The weight of $\beta \omega$ is $c$.
3. The space $\beta \omega$ is zero-dimensional.
4. Every infinite closed subset of $\beta \omega$ contains a topological copy of $\beta \omega$.
5. The space $\beta \omega$ contains no non-trivial convergent sequence.
6. No point of $\beta \omega-\omega$ is an isolated point.
7. The space $\beta \omega$ fails to have many properties involving the existence of non-trivial convergent sequence. For example:
$\text{ }$

1. The space $\beta \omega$ is not first countable at each point of the remainder $\beta \omega-\omega$.
2. The space $\beta \omega$ is not a Frechet space.
3. The space $\beta \omega$ is not a sequential space.
4. The space $\beta \omega$ is not sequentially compact.

$\text{ }$

8. No point of the remainder $\beta \omega-\omega$ is a $G_\delta$-point.
9. The remainder $\beta \omega-\omega$ does not have the countable chain condition. In fact, it has a disjoint open collection of cardinality $c$.

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Characterization Theorems

For any completely regular space $X$, let $C(X,I)$ be the set of all continuous functions from $X$ into $I=[0,1]$. The Stone-Cech compactification $\beta X$ is the subspace of the product space $[0,1]^{C(X,I)}$ which is the closure of the image of $X$ under the evaluation map $\beta:X \rightarrow [0,1]^{C(X,I)}$ (for the details, see Embedding Completely Regular Spaces into a Cube).

The brief sketch of $\beta \omega$ we present here is not based on the definition using the evaluation map. Instead we reply on some characterization theorems that are stated here (especially Theorem U3.1). These theorems uniquely describe the Stone-Cech compactification $\beta X$ of a given completely regular space $X$. For example, $\beta X$ satisfies the function extension property in Theorem C3 below. Furthermore any compactification $\alpha X$ of $X$ that satisfies the same property must be $\beta X$ (Theorem U3.1). So a “C” theorem tells us a property possessed by $\beta X$. The corresponding “U” theorem tells us that there is only one compactification (up to equivalence) that has this property.

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Theorem C1
Let $X$ be a completely regular space. Let $f:X \rightarrow Y$ be a continuous function from $X$ into a compact Hausdorff space $Y$. Then there is a continuous $F: \beta X \rightarrow Y$ such that $F \circ \beta=f$.

$\text{ }$

Theorem U1
If $K$ is any compactification of $X$ that satisfies condition in Theorem C1, then $K$ must be equivalent to $\beta X$.

See Two Characterizations of Stone-Cech Compactification.
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$\text{ }$

Theorem C2
Let $X$ be a completely regular space. Among all compactifications of the space $X$, the Stone-Cech compactification $\beta X$ of the space $X$ is maximal with respect to the partial order $\le$.

$\text{ }$

Theorem U2
The property in Theorem C2 is unique to $\beta X$. That is, if, among all compactifications of the space $X$, $\alpha X$ is maximal with respect to the partial order $\le$, then $\alpha X \approx \beta X$.

See Two Characterizations of Stone-Cech Compactification.
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$\text{ }$

Theorem C3
Let $X$ be a completely regular space. The space $X$ is $C^*$-embedded in its Stone-Cech compactification $\beta X$.

$\text{ }$

Theorem U3.1
Let $X$ be a completely regular space. Let $I=[0,1]$. Let $\alpha X$ be a compactification of $X$ such that each continuous $f:X \rightarrow I$ can be extended to a continuous $\hat{f}:\alpha X \rightarrow I$. Then $\alpha X$ must be $\beta X$.

$\text{ }$

Theorem U3.2
If $\alpha X$ is any compactification of $X$ that satisfies the property in Theorem C3 (i.e., $X$ is $C^*$-embedded in $\alpha X$), then $\alpha X$ must be $\beta X$.

See C*-Embedding Property and Stone-Cech Compactification.
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$\text{ }$

The following discussion illustrates how we can use some of these characterizations theorem to obtain information about $\beta X$ and $\beta \omega$ in particular.

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Result 1 and Result 2

According to the previous post (Stone-Cech Compactification is Maximal), we have for any completely regular space $X$, $\lvert \beta X \lvert \le 2^{2^{d(X)}}$ where $d(X)$ is the density (the smallest cardinality of a dense set in $X$). With $\omega$ being a countable space, $\lvert \beta \omega \lvert \le 2^{2^{\omega}}=2^c$.

Result 1 is established if we have $2^c \le \lvert \beta \omega \lvert$. Consider the cube $I^I$ where $I$ is the unit interval $I=[0,1]$. Since the product space of $c$ many separable space is separable (see Product of Separable Spaces), $I^I$ is separable. Let $S \subset I^I$ be a countable dense set. Let $f:\omega \rightarrow S$ be a bijection. Clearly $f$ is a continuous function from the discrete space $\omega$ into $I^I$. By Theorem C1, $f$ is extended by a continuous $F:\beta \omega \rightarrow I^I$. Note that the image $F(\beta \omega)$ is dense in $I^I$ since $F(\beta \omega)$ contains the dense set $S$. On the other hand, $F(\beta \omega)$ is compact. So $F(\beta \omega)=I^I$. Thus $F$ is a surjection. The cardinality of $I^I$ is $2^c$. Thus we have $2^c \le \lvert \beta \omega \lvert$.

From the same previous post (Stone-Cech Compactification is Maximal), it is shown that $w(\beta X) \le 2^{d(X)}$. Thus $w(\beta \omega) \le 2^{\omega}=c$. The same function $F:\beta \omega \rightarrow I^I$ in the above paragraph shows that $c \le w(\beta \omega)$ (see Lemma 2 in Stone-Cech Compactification is Maximal). Thus we have $w(\beta \omega)=c$ $\blacksquare$

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Result 3

A space is said to be zero-dimensional whenever it has a base consisting of open and closed sets. The proof that $\beta X$ is zero-dimensional comes after the following lemmas and theorems.

Theorem 1
Let $X$ be a normal space. If $H$ and $K$ are disjoint closed subsets of $X$, then $H$ and $K$ have disjoint closures in $\beta X$.

Proof of Theorem 1
Let $H$ and $K$ be disjoint closed subsets of $X$. By the normality of $X$ and by the Urysohn’s lemma, there is a continuous function $g:X \rightarrow [0,1]$ such that $g(H) \subset \left\{0 \right\}$ and $g(K) \subset \left\{1 \right\}$. By Theorem C3.1, $g$ can be extended by $G:\beta X \rightarrow [0,1]$. Note that $\overline{H} \subset G^{-1}(0)$ and $\overline{K} \subset G^{-1}(1)$. Thus $\overline{H} \cap \overline{K} = \varnothing$. $\blacksquare$

Theorem 2
Let $X$ be a completely regular space. Let $H$ be a closed and open subset of $X$. Then $\overline{H}$ (the closure of $H$ in $\beta X$) is also a closed and open set in $\beta X$.

Proof of Theorem 2
Let $H$ be a closed and open subset of $X$. Let $K=X-H$. Define $\gamma:X \rightarrow [0,1]$ by letting $\gamma(x)=0$ for all $x \in H$ and $\gamma(x)=1$ for all $x \in K$. Since both $H$ and $K$ are closed and open, the map $\gamma$ is continuous. By Theorem C3, $\gamma$ is extended by some continuous $\Gamma:\beta X \rightarrow [0,1]$. Note that $\overline{H} \subset \Gamma^{-1}(0)$ and $\overline{K} \subset \Gamma^{-1}(1)$. Thus $H$ and $K$ have disjoint closures in $\beta X$, i.e. $\overline{H} \cap \overline{K} = \varnothing$. Both $H$ and $K$ are closed and open in $\beta X$ since $\beta X=\overline{H} \cup \overline{K}$. $\blacksquare$

Lemma 3
For every $A \subset \omega$, $\overline{A}$ (the closure of $A$ in $\beta \omega$) is both closed and open in $\beta \omega$.

Note that Lemma 3 is a corollary of Theorem 2.

Lemma 4
Let $O \subset \beta \omega$ be a set that is both closed and open in $\beta \omega$. Then $O=\overline{A}$ where $A= O \cap \omega$.

Proof of Lemma 4
Let $A=O \cap \omega$. Either $O \subset \omega$ or $O \cap (\beta \omega-\omega) \ne \varnothing$. Thus $A \ne \varnothing$. We claim that $O=\overline{A}$. Since $A \subset O$, it follows that $\overline{A} \subset \overline{O}=O$. To show $O \subset \overline{A}$, pick $x \in O$. If $x \in \omega$, then $x \in A$. So focus on the case that $x \notin \omega$. It is clear that $x \notin \overline{B}$ where $B=\omega -A$. But every open set containing $x$ must contain some points of $\omega$. These points of $\omega$ must be points of $A$. Thus we have $x \in \overline{A}$. $\blacksquare$

Proof of Result 3
Let $\mathcal{A}$ be the set of all closed and open sets in $\beta \omega$. Let $\mathcal{B}=\left\{\overline{A}: A \subset \omega \right\}$. Lemma 3 shows that $\mathcal{B} \subset \mathcal{A}$. Lemma 4 shows that $\mathcal{A} \subset \mathcal{B}$. Thus $\mathcal{A}= \mathcal{B}$. We claim that $\mathcal{B}$ is a base for $\beta \omega$. To this end, we show that for each open $O \subset \beta \omega$ and for each $x \in O$, we can find $\overline{A} \in \mathcal{B}$ with $x \in \overline{A} \subset O$. Let $O$ be open and let $x \in O$. Since $\beta \omega$ is a regular space, we can find open set $V \subset \beta \omega$ with $x \in V \subset \overline{V} \subset O$. Let $A=V \cap \omega$.

We claim that $x \in \overline{A}$. Suppose $x \notin \overline{A}$. There exists open $U \subset V$ such that $x \in U$ and $U$ misses $\overline{A}$. But $U$ must meets some points of $\omega$, say, $y \in U \cap \omega$. Then $y \in V \cap \omega=A$, which is a contradiction. So we have $x \in \overline{A}$.

It is now clear that $x \in \overline{A} \subset \overline{V} \subset O$. Thus $\beta \omega$ is zero-dimensional since $\mathcal{B}$ is a base consisting of closed and open sets. $\blacksquare$

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Result 4 and Result 5

Result 5 is a corollary of Result 4. We first prove two lemmas before proving Result 4.

Lemma 5
For each infinite $A \subset \omega$, $\overline{A}$ (the closure of $A$ in $\beta \omega$) is a homeomorphic copy of $\beta \omega$ and thus has cardinality $2^c$.

Proof of Lemma 5
Let $A \subset \omega$. Let $g:A \rightarrow [0,1]$ be any function (necessarily continuous). Let $f:\omega \rightarrow [0,1]$ be defined by $f(x)=g(x)$ for all $x \in A$ and $f(x)=0$ for all $x \in \omega-A$. By Theorem C3, $f$ can be extended by $F:\beta \omega \rightarrow [0,1]$. Let $G=F \upharpoonright \overline{A}$.

Note that the function $G: \overline{A} \rightarrow [0,1]$ extends $g:A \rightarrow [0,1]$. Thus by Theorem U3.1, $\overline{A}$ must be $\beta A$. Since $A$ is a countably infinite discrete space, $\beta A$ must be equivalent to $\beta \omega$. $\blacksquare$

Lemma 6
For each countably infinite $A \subset \beta \omega-\omega$ such that $A$ is relatively discrete, $\overline{A}$ (the closure of $A$ in $\beta \omega$) is a homeomorphic copy of $\beta \omega$ and thus has cardinality $2^c$.

Proof of Lemma 6
Let $A=\left\{t_1,t_2,t_3,\cdots \right\} \subset \beta \omega -\omega$ such that $A$ is discrete in the relative topology inherited from $\beta \omega$. There exist disjoint open sets $G_1,G_2,G_3,\cdots$ (open in $\beta \omega$) such that for each $j$, $t_j \in G_j$. Since $\beta \omega$ is zero-dimensional (Result 3), $G_1,G_2,G_3,\cdots$ can be made closed and open.

Let $f:A \rightarrow [0,1]$ be a continuous function. We show that $f$ can be extended by $F:\overline{A} \rightarrow [0,1]$. Once this is shown, by Theorem U3.1, $\overline{A}$ must be $\beta A$. Since $A$ is a countable discrete space, $\beta A$ must be equivalent to $\beta \omega$.

We first define $w:\omega \rightarrow [0,1]$ by:

$\displaystyle w(n)=\left\{\begin{matrix}f(t_j)& \exists \ j \text{ such that } n \in \omega \cap G_j\\{0}&\text{otherwise} \end{matrix}\right.$

The function $w$ is well defined since each $n \in \omega$ is in at most one $G_j$. By Theorem C3, the function $w$ is extended by some continuous $W:\beta \omega \rightarrow [0,1]$. By Lemma 4, for each $j$, $G_j=\overline{\omega \cap G_j}$. Thus, for each $j$, $t_j \in \overline{\omega \cap G_j}$. Note that $W$ is a constant function on the set $\omega \cap G_j$ (mapping to the constant value of $f(t_j)$). Thus $W(t_j)=f(t_j)$ for each $j$. So let $F=W \upharpoonright \overline{A}$. Thus $F$ is the desired function that extends $f$. $\blacksquare$

Proof of Result 4
Let $C \subset \beta \omega$ be an infinite closed set. Either $C \cap \omega$ is infinite or $C \cap (\beta \omega-\omega)$ is infinite. If $C \cap \omega$ is infinite, then by Lemma 5, $\overline{C \cap \omega}$ is a homeomorphic copy of $\beta \omega$. Now focus on the case that $C_0=C \cap (\beta \omega-\omega)$ is infinite. We can choose inductively a countably infinite set $A \subset C_0$ such that $A$ is relatively discrete. Then by Lemma 6 $\overline{A}$ is a copy of $\beta \omega$ that is a subset of $C$. $\blacksquare$

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Result 6

We prove that no point in the remainder $\beta \omega-\omega$ is an isolated point. To see this, pick $x \in \beta \omega-\omega$ and pick an arbitrary closed and open set $O \subset \beta \omega$ with $x \in O$. Let $V=O \cap (\beta \omega-\omega)$ (thus an arbitrary open set in the remainder containing $x$). By Lemma 4, $O=\overline{A}$ where $A=O \cap \omega$. According to Lemma 5, $O=\overline{A}$ is a copy of $\beta \omega$ and thus has cardinality $2^c$. The set $V$ is $O$ minus a subset of $\omega$. Thus $V$ must contains $2^c$ many points. This means that $\left\{ x \right\}$ can never be open in the remainder $\beta \omega-\omega$. In fact, we just prove that any open and closed subset of $\beta \omega-\omega$ (thus any open subset) must have cardinality at least $2^c$. $\blacksquare$

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Result 7

The results under Result 7 are corollary of Result 5 (there is no non-trivial convergent sequence in $\beta \omega$). To see Result 7.1, note that every point $x$ in the remainder is not an isolated point and hence cannot have a countable local base (otherwise there would be a non-trivial convergent sequence converging to $x$).

A space $Y$ is said to be a Frechet space if $A \subset Y$ and for each $x \in \overline{A}$, there is a sequence $\left\{ x_n \right\}$ of points of $A$ such that $x_n \rightarrow x$. A set $A \subset Y$ is said to be sequentially closed in $Y$ if for any sequence $\left\{ x_n \right\}$ of points of $A$, $x_n \rightarrow x$ implies $x \in A$. A space $Y$ is said to be a sequential space if $A \subset Y$ is a closed set if and only if $A$ is a sequentially closed set. If a space is Frechet, then it is sequential. It is clear that $\beta \omega$ is not a sequential space.

A space is said to be sequentially compact if every sequence of points in this space has a convergent subsequence. Even though $\beta \omega$ is compact, it cannot be sequentially compact.

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Result 8

Result 7.1 indicates that no point of remainder $\beta \omega-\omega$ can have a countable local base. In fact, no point of the remainder can be a $G_\delta$-point (a point that is the intersection of countably many open sets). The remainder $\beta \omega-\omega$ is a compact space (being a closed subset of $\beta \omega$). In a compact space, if a point is a $G_\delta$-point, then there is a countable local base at that point (see 3.1.F (a) on page 135 of [1] or 17F.7 on page 125 of [2]). $\blacksquare$

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Result 9

The space $\beta \omega$ is a separable space since $\omega$ is a dense set. Thus $\beta \omega$ has the countable chain condition. However, the remainder $\beta \omega-\omega$ does not have the countable chain condition. We show that there is a disjoint collection of $c$ many open sets in $\beta \omega-\omega$.

There is a family $\mathcal{A}$ of infinite subsets of $\omega$ such that for every $A,B \in \mathcal{A}$ with $A \ne B$, $A \cap B$ is finite. Such a collection of sets is said to be an almost disjoint family. There is even an almost disjoint family of cardinality $c$ (see A Space with G-delta Diagonal that is not Submetrizable). Let $\mathcal{A}$ be such a almost disjoint family.

For each $A \in \mathcal{A}$, let $U_A=\overline{A}$ and $V_A=\overline{A} \cap (\beta \omega -\omega)$. By Lemma 3, each $U_A$ is a closed and open set in $\beta \omega$. Thus each $V_A$ is a closed and open set in the remainder $\beta \omega-\omega$. Note that $\left\{V_A: A \in \mathcal{A} \right\}$ is a disjoint collection of open sets in $\beta \omega-\omega$. $\blacksquare$

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Blog Posts on Stone-Cech Compactification

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Reference

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

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$\copyright \ \ 2012$

# Sorgenfrey Line is not a Moore Space

We found an incorrect statement about the Sorgenfrey line in an entry in Wikipedia about Moore space (link). This statement opens up a discussion on the question of whether the Sorgenfrey line is a Moore space as well as a discussion on Moore space. The following is the incorrect statement found in Wikipedia by the author.

The Sorgenfrey line is the space whose underlying set is the real line $S=\mathbb{R}$ where the topology is generated by a base consisting the half open intervals of the form $[a,b)$. The Sorgenfrey plane is the square $S \times S$.

Even though the Sorgenfrey line is normal, the Sorgenfrey plane is not normal. In fact, the Sorgenfrey line is the classic example of a normal space whose square is not normal. Both the Sorgenfrey line and the Sorgenfrey plane are not Moore space but not for the reason given. The statement seems to suggest that any normal Moore space is second countable. But this flies in the face of all the profound mathematics surrounding the normal Moore space conjecture, which is also discussed in the Wikipedia entry.

The statement indicated above is only a lead-in to a discussion of Moore space. We are certain that it will be corrected. We always appreciate readers who kindly alert us to errors found in this blog.

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Moore Spaces

Let $X$ be a regular space. A development for $X$ is a sequence $\mathcal{G}_1,\mathcal{G}_2,\mathcal{G}_3,\cdots$ of open covers of $X$ such that for each $x \in X$, and for each open subset $U$ of $X$ with $x \in U$, there exists one cover $\mathcal{G}_n$ satisfying the condition that for any open set $V \in \mathcal{G}_n$, $x \in V \Rightarrow V \subset U$. When $X$ has a development, $X$ is said to be a Moore space (also called developable space). A Note On The Sorgenfrey Line is an introductory note on the Sorgenfrey line.

Moore spaces can be viewed as a generalization of metrizable spaces. Moore spaces are first countable (having a countable base at each point). For a development $\mathcal{G}_1,\mathcal{G}_2,\mathcal{G}_3,\cdots$, the open sets in $\mathcal{G}_n$ are considered “smaller” as the index $n$ increases. In fact, this is how a development is defined for a metric space, where $\mathcal{G}_n$ consists of all open balls with diameters less than $\frac{1}{n}$. Thus metric spaces are developable. There are plenty of non-metrizable Moore space. One example is the Niemytzki’s Tangent Disc space.

In a Moore space, every closed set is a $G_\delta$-set. Thus if a Moore space is normal, it is perfectly normal. Any Moore space has a $G_\delta$-diagonal (the diagonal $\Delta=\left\{(x,x): x \in X \right\}$ is a $G_\delta$-set in $X \times X$). It is a well known theorem that every compact space with a $G_\delta$-diagonal is metrizable. Thus any compact Moore space is metrizable.

The last statement can be shown more directly. Suppose that $X$ is compact and has a development $\mathcal{G}_1,\mathcal{G}_2,\mathcal{G}_3,\cdots$. Then each $\mathcal{G}_n$ has a finite subcover $\mathcal{H}_n$. Then $\bigcup_{n=1}^\infty \mathcal{H}_n$ is a countable base for $X$. Thus any compact Moore space is second countable and hence metrizable.

What about paracompact Moore space? Suppose that $X$ is paracompact and has a development $\mathcal{G}_1,\mathcal{G}_2,\mathcal{G}_3,\cdots$. Then each $\mathcal{G}_n$ has a locally finite open refinement $\mathcal{H}_n$. Then $\bigcup_{n=1}^\infty \mathcal{H}_n$ is a $\sigma$-locally finite base for $X$. The Smirnov-Nagata metrization theorem states that a space is metrizable if and only if it has a $\sigma$-locally finite base (see Theorem 23.9 on page 170 of [2]). Thus any paracompact Moore space has a $\sigma$-locally finite base and is thus metrizable (after using the big gun of the Smirnov-Nagata metrization theorem).

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Sorgenfrey Line

The Sorgenfrey line is regular and Lindelof. Hence it is paracompact. Since the Sorgenfrey line is not metrizable, by the above discussion it cannot be a Moore space. The Sorgenfrey plane is also not a Moore space. Note that being a Moore space is a hereditary property. So if the Sorgenfrey plane is a Moore space, then every subspace of the Sorgenfrey plane (including the Sorgenfrey line) is a Moore space.

The following theorem is another way to show that the Sorgenfrey line is not a Moore space.

Bing’s Metrization Theorem
A topological space is metrizable if and only if it is a collectionwise normal Moore space.

Every paracompact space is collectionwise normal (see Theorem 5.1.18, p.305 of [1]). Thus the Sorgenfrey line is collectionwise normal and hence cannot be a Moore space. A space $X$ is said to be collectionwise normal if $X$ is a $T_1$-space and for every discrete collection $\left\{W_\alpha: \alpha \in A \right\}$ of closed sets in $X$, there exists a discrete collection $\left\{V_\alpha: \alpha \in A \right\}$ of open subsets of $X$ such that $W_\alpha \subset V_\alpha$. For a proof of Bing’s metrization theorem, see page 329 of [1].

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Remark

The normal Moore space conjecture is the statement that every normal Moore space is metrizable. This conjecture had been one of the key motivating questions for many set theorists and topologists during a large part of the twentieth century. The bottom line is that this statement cannot not be decided just on the basis of the set of generally accepted axioms called Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC. But Bing’s metrization theorem states that if we strengthen normality to collectionwise normality, we have a definite answer.

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Reference

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

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$\copyright \ \ 2012$

# Bernstein Sets Are Baire Spaces

A topological space $X$ is a Baire space if the intersection of any countable family of open and dense sets in $X$ is dense in $X$ (or equivalently, every nonempty open subset of $X$ is of second category in $X$). One version of the Baire category theorem implies that every complete metric space is a Baire space. The real line $\mathbb{R}$ with the usual Euclidean metric $\lvert x-y \lvert$ is a complete metric space, and hence is a Baire space. The space of irrational numbers $\mathbb{P}$ is also a complete metric space (not with the usual metric $\lvert x-y \lvert$ but with another suitable metric that generates the Euclidean topology on $\mathbb{P}$) and hence is also a Baire space. In this post, we show that there are subsets of the real line that are Baire space but not complete metric spaces. These sets are called Bernstein sets.

A Bernstein set, as discussed here, is a subset $B$ of the real line such that both $B$ and $\mathbb{R}-B$ intersect with every uncountable closed subset of the real line. We present an algorithm on how to generate such a set. Bernstein sets are not Lebesgue measurable. Our goal here is to show that Bernstein sets are Baire spaces but not weakly $\alpha$-favorable, and hence are spaces in which the Banach-Mazur game is undecidable.

Baire spaces are defined and discussed in this post. The Banach-Mazur game is discussed in this post. The algorithm of constructing Bernstein set is found in [2] (Theorem 5.3 in p. 23). Good references for basic terms are [1] and [3].
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In constructing Bernstein sets, we need the following lemmas.

Lemma 1
In the real line $\mathbb{R}$, any uncountable closed set has cardinality continuum.

Proof
In the real line, every uncountable subset of the real line has a limit point. In fact every uncountable subset of the real line contains at least one of its limit points (see The Lindelof property of the real line). Let $A \subset \mathbb{R}$ be an uncountable closed set. The set $A$ has to contain at least one of its limit point. As a result, at most countably many points of $A$ are not limit points of $A$. Take away these countably many points of $A$ that are not limit points of $A$ and call the remainder $A^*$. The set $A^*$ is still an uncountable closed set but with an additional property that every point of $A^*$ is a limit point of $A^*$. Such a set is called a perfect set. In Perfect sets and Cantor sets, II, we demonstrate a procedure for constructing a Cantor set out of any nonempty perfect set. Thus $A^*$ (and hence $A$) contains a Cantor set and has cardinality continuum. $\blacksquare$

Lemma 2
In the real line $\mathbb{R}$, there are continuum many uncountable closed subsets.

Proof
Let $\mathcal{B}$ be the set of all open intervals with rational endpoints, which is a countable set. The set $\mathcal{B}$ is a base for the usual topology on $\mathbb{R}$. Thus every nonempty open subset of the real line is the union of some subcollection of $\mathcal{B}$. So there are at most continuum many open sets in $\mathbb{R}$. Thus there are at most continuum many closed sets in $\mathbb{R}$. On the other hand, there are at least continuum many uncountable closed sets (e.g. $[-b,b]$ for $b \in \mathbb{R}$). Thus we can say that there are exactly continuum many uncountable closed subsets of the real line. $\blacksquare$

Constructing Bernstein Sets

Let $c$ denote the cardinality of the real line $\mathbb{R}$. By Lemma 2, there are only $c$ many uncountable closed subsets of the real line. So we can well order all uncountable closed subsets of $\mathbb{R}$ in a collection indexed by the ordinals less than $c$, say $\left\{F_\alpha: \alpha < c \right\}$. By Lemma 1, each $F_\alpha$ has cardinality $c$. Well order the real line $\mathbb{R}$. Let $\prec$ be this well ordering.

Based on the well ordering $\prec$, let $x_0$ and $y_0$ be the first two elements of $F_0$. Let $x_1$ and $y_1$ be the first two elements of $F_1$ (based on $\prec$) that are different from $x_0$ and $y_0$. Suppose that $\alpha < c$ and that for each $\beta < \alpha$, points $x_\beta$ and $y_\beta$ have been selected. Then $F_\alpha-\bigcup_{\beta<\alpha} \left\{x_\beta,y_\beta \right\}$ is nonempty since $F_\alpha$ has cardinality $c$ and only less than $c$ many points have been selected. Then let $x_\alpha$ and $y_\alpha$ be the first two points of $F_\alpha-\bigcup_{\beta<\alpha} \left\{x_\beta,y_\beta \right\}$ (according to $\prec$). Thus $x_\alpha$ and $y_\alpha$ can be chosen for each $\alpha.

Let $B=\left\{ x_\alpha: \alpha. Then $B$ is a Bernstein set. Note that $B$ meets every uncountable closed set $F_\alpha$ with the point $x_\alpha$ and the complement of $B$ meets every uncountable closed set $F_\alpha$ with the point $y_\alpha$.

The algorithm described here produces a unique Bernstein set that depends on the ordering of the uncountable closed sets $F_\alpha$ and the well ordering $\prec$ of $\mathbb{R}$.

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Key Lemmas

Baire spaces are defined and discussed in this previous post. Baire spaces can also be characterized using the Banach-Mazur game. The following lemmas establish that any Bernstein is a Baire space that is not weakly $\alpha$-favorable. Lemma 3 is applicable to all topological spaces. Lemmas 4, 5, 6, and 7 are specific to the real line.

Lemma 3
Let $Y$ be a topological space. Let $F \subset Y$ be a set of first category in $Y$. Then $Y-F$ contains a dense $G_\delta$ subset.

Proof
Let $F \subset Y$ be a set of first category in $Y$. Then $F=\bigcup \limits_{n=0}^\infty F_n$ where each $F_n$ is nowhere dense in $Y$. The set $X-\bigcup \limits_{n=0}^\infty \overline{F_n}$ is a dense $G_\delta$ set in the space $X$ and it is contained in the complement of $F$. We have:

$\displaystyle . \ \ \ \ \ X-\bigcup \limits_{n=0}^\infty \overline{F_n} \subset X-F$ $\blacksquare$

We now set up some notaions in preparation of proving Lemma 4 and Lemma 7. For any set $A \subset \mathbb{R}$, let $\text{int}(A)$ be the interior of the set $A$. Denote each positive integer $n$ by $n=\left\{0,1,\cdots,n-1 \right\}$. In particular, $2=\left\{0,1\right\}$. Let $2^{n}$ denote the collection of all functions $f: n \rightarrow 2$. Identify each $f \in 2^n$ by the sequence $f(0),f(1),\cdots,f(n-1)$. This identification makes notations in the proofs of Lemma 4 and Lemma 7 easier to follow. For example, for $f \in 2^n$, $I_f$ denotes a closed interval $I_{f(0),f(1),\cdots,f(n-1)}$. When we choose two disjoint subintervals of this interval, they are denoted by $I_{f,0}$ and $I_{f,1}$. For $f \in 2^n$, $f \upharpoonright 1$ refers to $f(0)$, $f \upharpoonright 2$ refers to the sequence $f(0),f(1)$, and $f \upharpoonright 3$ refers to the sequence $f(0),f(1),f(2)$ and so on.

The Greek letter $\omega$ denotes the first infinite ordinal. We equate it as the set of all nonnegative integers $\left\{0,1,2,\cdots \right\}$. Let $2^\omega$ denote the set of all functions from $\omega$ to $2=\left\{0,1 \right\}$.

Lemma 4
Let $W \subset \mathbb{R}$ be a dense $G_\delta$ set. Let $U$ be a nonempty open subset of $\mathbb{R}$. Then $W \cap U$ contains a Cantor set (hence an uncountable closed subset of the real line).

Proof
Let $W=\bigcap \limits_{n=0}^\infty O_n$ where each $O_n$ is an open and dense subset of $\mathbb{R}$. We describe how a Cantor set can be obtained from the open sets $O_n$. Take a closed interval $I_\varnothing=[a,b] \subset O_0 \cap U$. Let $C_0=I_\varnothing$. Then pick two disjoint closed intervals $I_{0} \subset O_1$ and $I_{1} \subset O_1$ such that they are subsets of the interior of $I_\varnothing$ and such that the lengths of both intervals are less than $2^{-1}$. Let $C_1=I_0 \cup I_1$.

At the $n^{th}$ step, suppose that all closed intervals $I_{f(0),f(1),\cdots,f(n-1)}$ (for all $f \in 2^n$) are chosen. For each such interval, we pick two disjoint closed intervals $I_{f,0}=I_{f(0),f(1),\cdots,f(n-1),0}$ and $I_{f,1}=I_{f(0),f(1),\cdots,f(n-1),1}$ such that each one is subset of $O_n$ and each one is subset of the interior of the previous closed interval $I_{f(0),f(1),\cdots,f(n-1)}$ and such that the lenght of each one is less than $2^{-n}$. Let $C_n$ be the union of $I_{f,0} \cup I_{f,1}$ over all $f \in 2^n$.

Then $C=\bigcap \limits_{j=0}^\infty C_j$ is a Cantor set that is contained in $W \cap U$. $\blacksquare$

Lemma 5
Let $X \subset \mathbb{R}$. If $X$ is not of second category in $\mathbb{R}$, then $\mathbb{R}-X$ contains an uncountable closed subset of $\mathbb{R}$.

Proof
Suppose $X$ is of first category in $\mathbb{R}$. By Lemma 3, the complement of $X$ contains a dense $G_\delta$ subset. By Lemma 4, the complement contains a Cantor set (hence an uncountable closed set). $\blacksquare$

Lemma 6
Let $X \subset \mathbb{R}$. If $X$ is not a Baire space, then $\mathbb{R}-X$ contains an uncountable closed subset of $\mathbb{R}$.

Proof
Suppose $X \subset \mathbb{R}$ is not a Baire space. Then there exists some open set $U \subset X$ such that $U$ is of first category in $X$. Let $U^*$ be an open subset of $\mathbb{R}$ such that $U^* \cap X=U$. We have $U=\bigcup \limits_{n=0}^\infty F_n$ where each $F_n$ is nowhere dense in $X$. It follows that each $F_n$ is nowhere dense in $\mathbb{R}$ too.

By Lemma 3, $\mathbb{R}-U$ contains $W$, a dense $G_\delta$ subset of $\mathbb{R}$. By Lemma 4, there is a Cantor set $C$ contained in $W \cap U^*$. This uncountable closed set $C$ is contained in $\mathbb{R}-X$. $\blacksquare$

Lemma 7
Let $X \subset \mathbb{R}$. Suppose that $X$ is a weakly $\alpha$-favorable space. If $X$ is dense in the open interval $(a,b)$, then there is an uncountable closed subset $C$ of $\mathbb{R}$ such that $C \subset X \cap (a,b)$.

Proof
Suppose $X$ is a weakly $\alpha$-favorable space. Let $\gamma$ be a winning strategy for player $\alpha$ in the Banach-Mazur game $BM(X,\beta)$. Let $(a,b)$ be an open interval in which $X$ is dense. We show that a Cantor set can be found inside $X \cap (a,b)$ by using the winning strategy $\gamma$.

Let $I_{-1}=[a,b]$. Let $t=b-a$. Let $U_{-1}^*=(a,b)$ and $U_{-1}=U^* \cap X$. We take $U_{-1}$ as the first move by the player $\beta$. Then the response made by $\alpha$ is $V_{-1}=\gamma(U_{-1})$. Let $C_{-1}=I_{-1}$.

Choose two disjoint closed intervals $I_0$ and $I_1$ that are subsets of the interior of $I_{-1}$ such that the lengths of these two intervals are less than $2^{-t}$ and such that $U_0^*=\text{int}(I_0)$ and $U_1^*=\text{int}(I_1)$ satisfy further properties, which are that $U_0=U_0^* \cap X \subset V_{-1}$ and $U_1=U_1^* \cap X \subset V_{-1}$ are open in $X$. Let $U_0$ and $U_1$ be two possible moves by player $\beta$ at the next stage. Then the two possible responses by $\alpha$ are $V_0=\gamma(U_{-1},U_0)$ and $V_1=\gamma(U_{-1},U_1)$. Let $C_1=I_0 \cup I_1$.

At the $n^{th}$ step, suppose that for each $f \in 2^n$, disjoint closed interval $I_f=I_{f(0),\cdots,f(n-1)}$ have been chosen. Then for each $f \in 2^n$, we choose two disjoint closed intervals $I_{f,0}$ and $I_{f,1}$, both subsets of the interior of $I_f$, such that the lengths are less than $2^{-(n+1) t}$, and:

• $U_{f,0}^*=\text{int}(I_{f,0})$ and $U_{f,1}^*=\text{int}(I_{f,1})$,
• $U_{f,0}=U_{f,0}^* \cap X$ and $U_{f,1}=U_{f,1}^* \cap X$ are open in $X$,
• $U_{f,0} \subset V_f$ and $U_{f,1} \subset V_f$

We take $U_{f,0}$ and $U_{f,1}$ as two possible new moves by player $\beta$ from the path $f \in 2^n$. Then let the following be the responses by player $\alpha$:

• $V_{f,0}=\gamma(U_{-1},U_{f \upharpoonright 1}, U_{f \upharpoonright 2}, \cdots,U_{f \upharpoonright (n-1)},U_f, U_{f,0})$
• $V_{f,1}=\gamma(U_{-1},U_{f \upharpoonright 1}, U_{f \upharpoonright 2}, \cdots,U_{f \upharpoonright (n-1)},U_f, U_{f,1})$

The remaining task in the $n^{th}$ induction step is to set $C_n=\bigcup \limits_{f \in 2^n} I_{f,0} \cup I_{f,1}$.

Let $C=\bigcap \limits_{n=-1}^\infty C_n$, which is a Cantor set, hence an uncountable subset of the real line. We claim that $C \subset X$.

Let $x \in C$. There there is some $g \in 2^\omega$ such that $\left\{ x \right\} = \bigcap \limits_{n=1}^\infty I_{g \upharpoonright n}$. The closed intervals $I_{g \upharpoonright n}$ are associated with a play of the Banach-Mazur game on $X$. Let the following sequence denote this play:

$\displaystyle (1) \ \ \ \ \ U_{-1},V_{-1},U_{g \upharpoonright 1},V_{g \upharpoonright 1},U_{g \upharpoonright 2},V_{g \upharpoonright 2},U_{g \upharpoonright 3},U_{g \upharpoonright 3}, \cdots$

Since the strategy $\gamma$ is a winning strategy for player $\alpha$, the intersection of the open sets in $(1)$ must be nonempty. Thus $\bigcap \limits_{n=1}^\infty V_{g \upharpoonright n} \ne \varnothing$.

Since the sets $V_{g \upharpoonright n} \subset I_{g \upharpoonright n}$, and since the lengths of $I_{g \upharpoonright n}$ go to zero, the intersection must have only one point, i.e., $\bigcap \limits_{n=1}^\infty V_{g \upharpoonright n} = \left\{ y \right\}$ for some $y \in X$. It also follows that $y=x$. Thus $x \in X$. We just completes the proof that $X$ contains an uncountable closed subset of the real line. $\blacksquare$

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Lemma 6 above establishes that any Bernstein set is a Baire space (if it isn’t, the complement would contain an uncountable closed set). Lemma 7 establishes that any Bernstein set is a topological space in which the player $\alpha$ has no winning strategy in the Banach-Mazur game (if player $\alpha$ always wins in a Bernstein set, it would contain an uncountable closed set). Thus any Bernstein set cannot be a weakly $\alpha$ favorable space. According to this previous post about the Banach-Mazur game, Baire spaces are characterized as the spaces in which the player $\beta$ has no winning strategy in the Banach-Mazur game. Thus any Bernstein set in a topological space in which the Banach-Mazur game is undecidable (i.e. both players in the Banach-Mazur game have no winning strategy).

One interesting observation about Lemma 6 and Lemma 7. Lemma 6 (as well as Lemma 5) indicates that the complement of a “thin” set contains a Cantor set. On the other hand, Lemma 7 indicates that a “thick” set contains a Cantor set (if it is dense in some open interval).

Reference

1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
2. Oxtoby, J. C., Measure and Category, Graduate Texts in Mathematics, Springer-Verlag, New York, 1971.
3. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

# The Banach-Mazur Game

A topological space $X$ is said to be a Baire space if for every countable family $\left\{U_0,U_1,U_2,\cdots \right\}$ of open and dense subsets of $X$, the intersection $\bigcap \limits_{n=0}^\infty U_n$ is dense in $X$ (equivalently if every nonempty open subset of $X$ is of second category in $X$). By the Baire category theorem, every complete metric space is a Baire space. The Baire property (i.e. being a Baire space) can be characterized using the Banach-Mazur game, which is the focus of this post.

Baire category theorem and Baire spaces are discussed in this previous post. We define the Banach-Mazur game and show how this game is related to the Baire property. We also define some completeness properties stronger than the Baire property using this game. For a survey on Baire spaces, see [4]. For more information about the Banach-Mazur game, see [1]. Good references for basic topological terms are [3] and [5]. All topological spaces are assumed to be at least Hausdorff.

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The Banach-Mazur Game

The Banach-Mazur game is a two-person game played on a topological space. Let $X$ be a space. There are two players, $\alpha$ and $\beta$. They take turn choosing nested decreasing nonempty open subsets of $X$ as follows. The player $\beta$ goes first by choosing a nonempty open subset $U_0$ of $X$. The player $\alpha$ then chooses a nonempty open subset $V_0 \subset U_0$. At the nth play where $n \ge 1$, $\beta$ chooses an open set $U_n \subset V_{n-1}$ and $\alpha$ chooses an open set $V_n \subset U_n$. The player $\alpha$ wins if $\bigcap \limits_{n=0}^\infty V_n \ne \varnothing$. Otherwise the player $\beta$ wins.

If the players in the game described above make the moves $U_0,V_0,U_1,V_1,U_2,V_2,\cdots$, then this sequence of open sets is said to be a play of the game.

The Banach-Mazur game, as described above, is denoted by $BM(X,\beta)$. In this game, the player $\beta$ makes the first move. If we modify the game by letting $\alpha$ making the first move, we denote this new game by $BM(X,\alpha)$. In either version, the goal of player $\beta$ is to reach an empty intersection of the chosen open sets while player $\alpha$ wants the chosen open sets to have nonempty intersection.

Before relating the Banach-Mazur game to Baire spaces, we give a remark about topological games. For any two-person game played on a topological space, we are interested in the following question.

• Can a player, by making his/her moves judiciously, insure that he/she will always win no matter what moves the other player makes?

If the answer to this question is yes, then the player in question is said to have a winning strategy. For an illustration, consider a space $X$ that is of first category in itself, so that $X=\bigcup \limits_{n=0}^\infty X_n$ where each $X_n$ is nowhere dense in $X$. Then player $\beta$ has a winning strategy in the Banach-Mazur game $BM(X,\beta)$. The player $\beta$ always wins the game by making his/her nth play $U_n \subset V_{n-1} - \overline{X_n}$.

In general, a strategy for a player in a game is a rule that specifies what moves he/she will make in every possible situation. In other words, a strategy for a player is a function whose domain is the set of all partial plays of the game, and this function tells the player what the next move should be. A winning strategy for a player is a strategy such that this player always wins if that player makes his/her moves using this strategy. A strategy for a player in a game is not a winning strategy if of all the plays of the game resulting from using this strategy, there is at least one specific play of the game resulting in a win for the other player.

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Strategies in the Banach-Mazur Game

With the above discussion in mind, let us discuss the strategies in the Banach-Mazur game. We show that the strategies in this game code a great amount of information about the topological space in which the game is played.

First we discuss strategies for player $\beta$ in the game $BM(X,\beta)$. A strategy for player $\beta$ is a function $\sigma$ such that $U_0=\sigma(\varnothing)$ (the first move) and for each partial play of the game ($n \ge 1$)

$\displaystyle (*) \ \ \ \ \ \ U_0,V_0,U_1,V_1,\cdots,U_{n-1},V_{n-1}$,

$U_n=\sigma(U_0,V_0,U_1,V_1,\cdots,U_{n-1},V_{n-1})$ is a nonempty open set such that $U_n \subset V_{n-1}$. If player $\beta$ makes all his/her moves using the strategy $\sigma$, then the strategy $\sigma$ for player $\beta$ contains information on all moves of $\beta$. We adopt the convention that a strategy for a player in a game depends only on the moves of the other player. Thus for the partial play of the Banach-Mazur game denoted by $(*)$ above, $U_n=\sigma(V_0,V_1,\cdots,V_{n-1})$.

If $\sigma$ is a winning strategy for player $\beta$ in the game $BM(X,\beta)$, then using this strategy will always result in a win for $\beta$. On the other hand, if $\sigma$ is a not a winning strategy for player $\beta$ in the game $BM(X,\beta)$, then there exists a specific play of the Banach-Mazur game

$\displaystyle . \ \ \ \ \ \ U_0,V_0,U_1,V_1,\cdots,U_{n-1},V_{n-1},\cdots$

such that $U_0=\sigma(\varnothing)$, and for each $n \ge 1$, $U_n=\sigma(V_0,\cdots,V_{n-1})$ and player $\alpha$ wins in this play of the game, that is, $\bigcap \limits_{n=0}^\infty V_n \ne \varnothing$.

In the game $BM(X,\alpha)$ (player $\alpha$ making the first move), a strategy for player $\beta$ is a function $\gamma$ such that for each partial play of the game

$\displaystyle (**) \ \ \ \ \ V_0,U_1,V_1,\cdots,U_{n-1},V_{n-1}$,

$U_n=\gamma(V_0,V_1,\cdots,V_{n-1})$ is a nonempty open set such that $U_n \subset V_{n-1}$.

We now present a lemma that helps translate game information into topological information.

Lemma 1
Let $X$ be a space. Let $O \subset X$ be a nonempty open set. Let $\tau$ be the set of all nonempty open subsets of $O$. Let $f: \tau \longrightarrow \tau$ be a function such that for each $V \in \tau$, $f(V) \subset V$. Then there exists a disjoint collection $\mathcal{U}$ consisting of elements of $f(\tau)$ such that $\bigcup \mathcal{U}$ is dense in $O$.

Proof
This is an argument using Zorn’s lemma. If the open set $O$ in the hypothesis has only one point, then the conclusion of the lemma holds. So assume that $O$ has at least two points.

Let $\mathcal{P}$ be the set consisting of all collections $\mathcal{F}$ such that each $\mathcal{F}$ is a disjoint collection consisting of elements of $f(\tau)$. First $\mathcal{P} \ne \varnothing$. To see this, let $V$ and $W$ be two disjoint open sets such that $V \subset O$ and $W \subset O$. This is possible since $O$ has at least two points. Let $\mathcal{F^*}=\left\{ f(V),f(W)\right\}$. Then we have $\mathcal{F^*} \in \mathcal{P}$. Order $\mathcal{P}$ by set inclusion. It is straightforward to show that $(\mathcal{P}, \subset)$ is a partially ordered set.

Let $\mathcal{T} \subset \mathcal{P}$ be a chain (a totally ordered set). We wish to show that $\mathcal{T}$ has an upper bound in $\mathcal{P}$. The candidate for an upper bound is $\bigcup \mathcal{T}$ since it is clear that for each $\mathcal{F} \in \mathcal{T}$, $\mathcal{F} \subset \bigcup \mathcal{T}$. We only need to show $\bigcup \mathcal{T} \in \mathcal{P}$. To this end, we need to show that any two elements of $\bigcup \mathcal{T}$ are disjoint open sets.

Note that elements of $\bigcup \mathcal{T}$ are elements of $f(\tau)$. Let $T_1,T_2 \in \bigcup \mathcal{T}$. Then $T_1 \in \mathcal{F}_1$ and $T_2 \in \mathcal{F}_2$ for some $\mathcal{F}_1 \in \mathcal{T}$ and $\mathcal{F}_2 \in \mathcal{T}$. Since $\mathcal{T}$ is a chain, either $\mathcal{F}_1 \subset \mathcal{F}_2$ or $\mathcal{F}_2 \subset \mathcal{F}_1$. This means that $T_1$ and $T_2$ belong to the same disjoing collection in $\mathcal{T}$. So they are disjoint open sets that are members of $f(\tau)$.

By Zorn’s lemma, $(\mathcal{P}, \subset)$ has a maximal element $\mathcal{U}$, which is a desired disjoint collection of sets in $f(\tau)$. Since $\mathcal{U}$ is maximal with respect to $\subset$, $\bigcup \mathcal{U}$ is dense in $O$. $\blacksquare$

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Characterizing Baire Spaces using the Banach-Mazur Game

Lemma 1 is the linkage between the Baire property and the strategies in the Banach-Mazur game. The thickness in Baire spaces and spaces of second category allow us to extract a losing play in any strategy for player $\beta$. The proofs for both Theorem 1 and Theorem 2 are very similar (after adjusting for differences in who makes the first move). Thus we only present the proof for Theorem 1.

Theorem 1
The space $X$ is a Baire space if and only if player $\beta$ has no winning strategy in the game $BM(X,\beta)$.

Proof
$\Longleftarrow$ Suppose that $X$ is not a Baire space. We define a winning strategy in the game $BM(X,\beta)$ for player $\beta$. The space $X$ not being a Baire space implies that there is some nonempty open set $U_0 \subset X$ such that $U_0$ is of first category in $X$. Thus $U_0=\bigcup \limits_{n=1}^\infty F_n$ where each $F_n$ is nowhere dense in $X$.

We now define a winning strategy for $\beta$. Let $U_0$ be the first move of $\beta$. For each $n \ge 1$, let player $\beta$ make his/her move by letting $U_n \subset V_{n-1} - \overline{F_n}$ if $V_{n-1}$ is the last move by $\alpha$. It is clear that whenever $\beta$ chooses his/her moves in this way, the intersection of the open sets has to be empty.

$\Longrightarrow$ Suppose that $X$ is a Baire space. Let $\sigma$ be a strategy for the player $\beta$. We show that $\sigma$ cannot be a winning strategy for $\beta$.

Let $U_0=\sigma(\varnothing)$ be the first move for $\beta$. For each open $V_0 \subset U_0$, $\sigma(V_0) \subset V_0$. Apply Lemma 1 to obtain a disjoint collection $\mathcal{U}_0$ consisting of open sets of the form $\sigma(V_0)$ such that $\bigcup \mathcal{U}_0$ is dense in $U_0$.

For each $W=\sigma(V_0) \in \mathcal{U}_0$, we have $\sigma(V_0,V_1) \subset V_1$ for all open $V_1 \subset W$. So the function $\sigma(V_0,\cdot)$ is like the function $f$ in Lemma 1. We can then apply Lemma 1 to obtain a disjoint collection $\mathcal{U}_1(W)$ consisting of open sets of the form $\sigma(V_0,V_1)$ such that $\bigcup \mathcal{U}_1(W)$ is dense in $W$. Then let $\mathcal{U}_1=\bigcup_{W \in \mathcal{U}_0} \mathcal{U}_1(W)$. Based on how $\mathcal{U}_1(W)$ are obtained, it follows that $\bigcup \mathcal{U}_1$ is dense in $U_0$.

Continue the inductive process in the same manner, we can obtain, for each $n \ge 1$, a disjoint collection $\mathcal{U}_n$ consisting of open sets of the form $\sigma(V_0,\dots,V_{n-1})$ (these are moves made by $\beta$ using the strategy $\sigma$) such that $\bigcup \mathcal{U}_n$ is dense in $U_0$.

For each $n$, let $O_n=\bigcup \mathcal{U}_n$. Each $O_n$ is dense open in $U_0$. Since $X$ is a Baire space, every nonempty open subset of $X$ is of second category in $X$ (including $U_0$). Thus $\bigcap \limits_{n=0}^\infty O_n \ne \varnothing$. From this nonempty intersection, we can extract a play of the game such that the open sets in this play of the game have one point in common (i.e. player $\alpha$ wins). We can extract the play of the game because the collection $\mathcal{U}_n$ are disjoint. Thus the strategy $\sigma$ is not a winning strategy for $\beta$. This completes the proof of Theorem 1. $\blacksquare$

Theorem 2
The space $X$ is of second category in itself if and only if player $\beta$ has no winning strategy in the game $BM(X,\alpha)$.

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Some Completeness Properties

Theorem 1 shows that a Baire space is one in which the player $\beta$ has no winning strategy in the Banach-Mazur game (the version in which $\beta$ makes the first move). In such a space, no matter what strategy player $\beta$ wants to use, it can be foiled by player $\alpha$ by producing one specific play in which $\beta$ loses. We now consider spaces in which player $\alpha$ has a winning strategy. A space $X$ is said to be a weakly $\alpha$-favorable if player $\alpha$ has a winning strategy in the game $BM(X,\beta)$. If $\alpha$ always wins, then $\beta$ has no winning strategy. Thus the property of being a weakly $\alpha$-favorable space is stronger than the Baire property.

In any complete metric space, the player $\alpha$ always has a winning strategy. The same idea used in proving the Baire category theorem can be used to establish this fact. By playing the game in a complete metric space, player $\alpha$ can ensure a win by making sure that the closure of his/her moves have diameters converge to zero (and the closure of his/her moves are subsets of the previous moves).

Based on Theorem 1, any Baire space is a space in which player $\beta$ of the Banach-Mazur game has no winning strategy. Any Baire space that is not weakly $\alpha$-favorable is a space in which both players of the Banach-Mazur game have no winning strategy (i.e. the game is undecidable). Any subset of the real line $\mathbb{R}$ that is a Bernstein set is such a space. A subset $B$ of the real line is said to be a Bernstein set if $B$ and its complement intersect every uncountable closed subset of the real line. Bernstein sets are discussed here.

Suppose $\theta$ is a strategy for $\alpha$ in the game $BM(X,\beta)$. If at each step, the strategy $\theta$ can provide a move based only on the other player’s last move, it is said to be a stationary strategy. For example, in the partial play $U_0,V_0,\cdots,U_{n-1},V_{n-1},U_n$, the strategy $\theta$ can determine the next move for $\alpha$ by only knowing the last move of $\beta$, i.e., $V_n=\theta(U_n)$. A space $X$ is said to be $\alpha$-favorable if player $\alpha$ has a stationary winning strategy in the game $BM(X,\beta)$. Clearly, any $\alpha$-favorable spaces are weakly $\alpha$-favorable spaces. However, there are spaces in which player $\alpha$ has a winning strategy in the Banach-Mazur game and yet has no stationary winning strategy (see [2]). Stationary winning strategy for $\alpha$ is also called $\alpha$-winning tactic (see [1]).

Reference

1. Choquet, G., Lectures on analysis, Vol I, Benjamin, New York and Amsterdam, 1969.
2. Deb, G., Stategies gagnantes dans certains jeux topologiques, Fund. Math. 126 (1985), 93-105.
3. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
4. Haworth, R. C., McCoy, R. A., Baire Spaces, Dissertations Math., 141 (1977), 1 – 73.
5. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.

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Revised 4/4/2014. $\copyright \ 2014 \text{ by Dan Ma}$