A theorem about CCC spaces

It is a well known result in general topology that in any regular space with the countable chain condition, paracompactness and the Lindelof property are equivalent. The proof of this result hinges on one theorem about the spaces with the countable chain condition. In this post we are to put the spotlight on this theorem (Theorem 1 below) and then use it to prove a few results. These results indicate that in a space with the countable chain condition with some weaker covering property is either Lindelof or paracompact.

This post is centered on a theorem about the CCC property (Theorem 1 and Theorem 1a below). So it can be considered as a continuation of a previous post on CCC called Some basic properties of spaces with countable chain condition. The results that are derived from Theorem 1 are also found in [2]. But the theorem concerning CCC is only a small part of that paper among several other focuses. In this post, the exposition is to explain several interesting theorems that are derived from Theorem 1. One of the theorems is the statement that every locally compact metacompact perfectly normal space is paracompact, a theorem originally proved by Arhangelskii (see Theorem 11 below).

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CCC Spaces

All spaces under consideration are at least T_1 and regular. A space X is said to have the countable chain condition (to have the CCC for short) if \mathcal{U} is a disjoint collection of non-empty open subsets of X (meaning that for any A,B \in \mathcal{U} with A \ne B, we have A \cap B=\varnothing), then \mathcal{U} is countable. In other words, in a space with the CCC, there cannot be uncountably many pairwise disjoint non-empty open sets. For ease of making a statement or stating a result, if X has the CCC, we also say that X is a CCC space or X is CCC.

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A Theorem about CCC Spaces

The theorem of CCC spaces we want to discuss has to do with collections of open sets that are “nice”. We first define what we mean by nice. Let \mathcal{A} be a collection of non-empty subsets of the space X. The collection \mathcal{A} is said to be point-finite (point-countable) if each point of X belongs to only finitely (countably) many sets in \mathcal{A}.

Now we define what we mean by “nice” collection of open sets. The collection \mathcal{A} is said to be locally finite (locally countable) at a point x \in X if there exists an open set O \subset X with x \in O such that O meets at most finitely (countably) many sets in \mathcal{A}. The collection \mathcal{A} is said to be locally finite (locally countable) if it is locally finite (locally countable) at each x \in X.

The property of being a separable space implies the CCC. The reverse is not true. However the CCC property is still a very strong property. The CCC property is equivalent to the property that if a collection of non-empty open sets is “nice” on a dense set of points, then the collection of open sets is a countable collection. The following is a precise statement.

    Theorem 1

      Let X be a CCC space. Then if \mathcal{U} is a collection of non-empty open subsets of X such that the following set

        D(\mathcal{U})=\left\{x \in X: \mathcal{U} \text{ is locally-countable at } x \right\}

      is dense in the open subspace \bigcup \mathcal{U}, then \mathcal{U} must be countable.

The collections of open sets in the above theorem do not have to be open covers. However, if they are open covers, the theorem can tie CCC spaces with some covering properties. As long as the space has the CCC, any open cover that is locally-countable on a dense set must be countable. Looking at it in the contrapositive angle, in a CCC space, any uncountable open cover is not locally-countable in some open set.

Proof of Theorem 1
Let \mathcal{U} be a collection of open subsets of X such that the set D(\mathcal{U}) as defined above is dense in the open subspace \bigcup \mathcal{U}. We show that \mathcal{U} is countable. Suppose not.

For each U \in \mathcal{U}, since U \cap D(\mathcal{U}) \ne \varnothing, we can choose a non-empty open set f(U) \subset U such that f(U) has non-empty intersection with only countably many sets in \mathcal{U}. Let \mathcal{U}_f be the following collection:

    \mathcal{U}_f=\left\{f(U): U \in \mathcal{U} \right\}

For H,K \in \mathcal{U}_f, by a chain from H to K, we mean a finite collection

    \left\{W_1,W_2,\cdots,W_n \right\} \subset \mathcal{U}_f

such that H=W_1, K=W_n and W_j \cap W_{j+1} \ne \varnothing for any 1 \le j <n. For each open set W \in \mathcal{U}_f, define \mathcal{C}(W) and \mathcal{E}(W) as follows:

    \mathcal{C}(W)=\left\{V \in \mathcal{U}_f: \text{there exists a chain from } W \text{ to } V \right\}

    \mathcal{E}(W)=\bigcup \mathcal{C}(W)

One observation we make is that for W_1,W_2 \in \mathcal{U}_f, if \mathcal{E}(W_1) \cap \mathcal{E}(W_2) \ne \varnothing, then \mathcal{C}(W_1)=\mathcal{C}(W_2) and \mathcal{E}(W_1)=\mathcal{E}(W_2). So the distinct \mathcal{E}(W) are pairwise disjoint. Because the space X has the CCC, there can be only countably many distinct open sets \mathcal{E}(W). Thus there can be only countably many distinct collections \mathcal{C}(W).

Note that each \mathcal{C}(W) is a countable collection of open sets. Each V \in \mathcal{U}_f meets only countably many open sets in \mathcal{U}. So each V \in \mathcal{U}_f can meet only countably many sets in \mathcal{U}_f, since for each V \in \mathcal{U}_f, V \subset U for some U \in \mathcal{U}. Thus for each W \in \mathcal{U}_f, in considering all finite-length chain starting from W, there can be only countably many open sets in \mathcal{U}_f that can be linked to W. Thus \mathcal{C}(W) must be countable. In taking the union of all \mathcal{C}(W), we get back the collection \mathcal{U}_f. Thus we have:

    \mathcal{U}_f=\bigcup \limits_{W \in \mathcal{U}_f} \mathcal{C}(W)

Because the space X is CCC, there are only countably many distinct collections \mathcal{C}(W) in the above union. Each \mathcal{C}(W) is countable. So \mathcal{U}_f is a countable collection of open sets.

Furthermore, each U \in \mathcal{U} contains at least one set in \mathcal{U}_f. From the way we choose sets in \mathcal{U}_f, we see that for each V \in \mathcal{U}_f, V=f(U) \subset U for at most countably many U \in \mathcal{U}. The argument indicates that we have a one-to-countable mapping from \mathcal{U}_f to \mathcal{U}. Thus the original collection \mathcal{U} must be countable. \blacksquare

The property in Theorem 1 is actually equivalent to the CCC property. Just that the proof of Theorem 1 represents the hard direction that needs to be proved. Theorem 1 can be expanded to be the following theorem.

    Theorem 1a

      Let X be a space. Then the following conditions are equivalent.

      1. The space X has the CCC.
      2. If \mathcal{U} is a collection of non-empty open subsets of X such that the following set

          D(\mathcal{U})=\left\{x \in X: \mathcal{U} \text{ is locally-countable at } x \right\}

        is dense in the open subspace \bigcup \mathcal{U}, then \mathcal{U} must be countable.

      3. If \mathcal{U} is a collection of non-empty open subsets of X such that \mathcal{U} is locally-countable at every point in the open subspace \bigcup \mathcal{U}, then \mathcal{U} must be countable.

The direction 1 \rightarrow 2 has been proved above. The directions 2 \rightarrow 3 and 3 \rightarrow 1 are straightforward.

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Tying Theorem 1 to “Nice” Open Covers

One easy application of Theorem 1 is to tie it to locally-finite and locally-countable open covers. We have the following theorem.

    Theorem 2

      In any CCC space, any locally-countable open cover must be countable. Thus any locally-finite open cover must also be countable.

Theorem 2 gives the well known result that any CCC paracompact space is Lindelof (see Theorem 5 below). In fact, Theorem 2 gives the result that any CCC para-Lindelof space is Lindelof (see Theorem 6 below). A space X is para-Lindelof if every open cover has a locally-countable open refinement.

Can Theorem 2 hold for point-finite covers (or point-countable covers)? The answer is no (see Example 1 below). With the additional property of having a Baire space, we have the following theorem.

    Theorem 3

      In any Baire space with the CCC, any point-finite open cover must be countable.

A Space X is a Baire space if U_1,U_2,U_3,\cdots are dense open subsets of X, then \bigcap \limits_{j=1}^\infty U_j \ne \varnothing. For more information about Baire spaces, see this previous post.
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Proof of Theorem 3
Let X be a Baire space with the CCC. Let \mathcal{U} be a point-finite open cover of X. Suppose that \mathcal{U} is uncountable. We show that this assumption with lead to a contradiction. Thus \mathcal{U} must be countable.

By Theorem 1, there exists an open set V \subset X such that \mathcal{U} is not locally-countable at any point in V. For each positive integer n, let H_n be the following:

    H_n=\left\{x \in V: x \text{ is in at most } n \text{ sets in } \mathcal{U} \right\}

Note that V=\bigcup \limits_{j=1}^\infty H_j. Furthermore, each H_n is a closed set in the space V. Since X is a Baire space, every non-empty open subset of X is of second category (i.e. it cannot be a union of countably many closed and nowhere dense sets). Thus it cannot be that each H_n is nowhere dense in V. For some n, H_n is not nowhere dense. There must exist some open W \subset V such that H_n \cap W is dense in W. Because H_n is closed, W \subset H_n.

Choose y \in W. The point y is in at most n open sets in \mathcal{U}. Let U_1,U_2,\cdots,U_m \in \mathcal{U} such that y \in \bigcap \limits_{j=1}^m U_j. Clearly 1 \le m \le n. Let U=W \cap U_1 \cap \cdots \cap U_m. Note that y \in U \subset H_n \subset V.

Every point in U belongs to at most n many sets in \mathcal{U} and already belong to m sets in \mathcal{U}. So each point in U can belong to at most n-m additional open sets in \mathcal{U}. Consider the case n-m=0 and the case n-m>0. We show that each case leads to a contradiction.

Suppose that n-m=0. Then each point of U can only meet n open sets in \mathcal{U}, namely U_1,U_2,\cdots,U_m. This contradicts that \mathcal{U} is not locally-countable at points in U \subset V.

Suppose that k=n-m>0. Let \mathcal{U}^*=\mathcal{U}-\left\{U_1,\cdots,U_m \right\}. Let \mathcal{M} be the following collection:

    \mathcal{M}=\left\{U \cap \bigcap \limits_{O \in M} O \ne \varnothing: M \subset \mathcal{U}^* \text{ and } \lvert M \lvert=k \right\}

Each element of \mathcal{M} is an open subset of U that is the intersection of exactly n many open sets in \mathcal{U}. So \mathcal{M} is a collection of pairwise disjoint open sets. The open set U as a topological space has the CCC. So \mathcal{M} is at most countable. Thus the open set U meets at most countably many open sets in \mathcal{U}, contradicting that \mathcal{U} is not locally-countable at points in U \subset V.

Both cases n-m=0 and n-m>0 lead to contradiction. So \mathcal{U} must be countable. The proof to Theorem 3 is completed. \blacksquare

As a corollary to Theorem 3, we have the result that every Baire CCC metacompact space is Lindelof.

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Some Applications of Theorems 2 and 3

In proving paracompactness in some of the theorems, we need a theorem involving the concept of star-countable open cover. A collection \mathcal{A} of subsets of a space X is said to be star-finite (star-countable) if for each A \in \mathcal{A}, only finitely (countably) many sets in \mathcal{A} meets A, i.e., the following set

    \left\{B \in \mathcal{A}: B \cap A \ne \varnothing \right\}

is finite (countable). The proof of the following theorem can be found in Engleking (see the direction (iv) implies (i) in the proof of Theorem 5.3.10 on page 326 in [1]).

    Theorem 4

      If every open cover of a regular space X has a star-countable open refinement, then X is paracompact.

As indicated in the above section, Theorem 2 and Theorem 3 have some obvious applications. We have the following theorems.

    Theorem 5

      Let X be a CCC space. Then X is paracompact if and only of X is Lindelof.

Proof of Theorem 5
The direction \Longleftarrow follows from the fact that any regular Lindelof space is paracompact.

The direction \Longrightarrow follows from Theorem 2. \blacksquare

    Theorem 6

      Every CCC para-Lindelof space is Lindelof.

Proof of Theorem 6
This also follows from Theorem 2. \blacksquare

    Theorem 7

      Every Baire CCC metacompact space is Lindelof.

Proof of Theorem 7
Let X be a Baire CCC metacompact space. Let \mathcal{U} be an open cover of X. By metacompactness, let \mathcal{V} be a point-finite open refinement of \mathcal{U}. By Theorem 3, \mathcal{V} must be countable. \blacksquare

    Theorem 8

      Every Baire CCC hereditarily metacompact space is hereditarily Lindelof.

Proof of Theorem 8
Let X be a Baire CCC hereditarily metacompact space. To show that X is hereditarily Lindelof, it suffices to show that every non-empty open subset is Lindelof. Let Y \subset X be open. Then Y has the CCC and is also metacompact. Being a Baire space is hereditary with respect to open subspaces. So Y is a Baire space too. By Theorem 7, Y is Lindelof. \blacksquare

    Theorem 9

      Every locally CCC regular para-Lindelof space is paracompact.

Proof of Theorem 9
A space is locally CCC if every point has an open neighborhood that has the CCC. Let X be a regular space that is locally CCC and para-Lindelof. Let \mathcal{U} be an open cover of X. Using the locally CCC assumption and by taking a refinement of \mathcal{U} if necessary, we can assume that each open set in \mathcal{U} has the CCC. By the para-Lindelof assumption, let \mathcal{V} be a locally-countable open refinement of \mathcal{U}. So each open set in \mathcal{V} has the CCC too.

Now we show that \mathcal{V} is star-countable. Let V \in \mathcal{V}. Let \mathcal{G} be the following collection:

    \mathcal{G}=\left\{V \cap W: W \in \mathcal{V} \right\}

which is is open cover of V. Within the subspace V, \mathcal{G} is a locally-countable open cover. By Theorem 2, \mathcal{G} must be countable. The collection \mathcal{G} represents all the open sets in \mathcal{V} that have non-empty intersection with V. Thus only countably many open sets in \mathcal{V} can meet V. So \mathcal{V} is a star-countable open refinement of \mathcal{U}. By Theorem 4, X is paracompact. \blacksquare

    Theorem 10

      Every locally CCC regular metacompact Baire space is paracompact.

Proof of Theorem 10
Let X be a regular space that is locally CCC and is a metacompact Baire space. Let \mathcal{U} be an open cover of X. Using the locally CCC assumption and by taking a refinement of \mathcal{U} if necessary, we can assume that each open set in \mathcal{U} has the CCC. By the metacompact assumption, let \mathcal{V} be a point-finite open refinement of \mathcal{U}. So each open set in \mathcal{V} has the CCC too. Each open set in \mathcal{V} is also a Baire space.

Now we show that \mathcal{V} is star-countable. Let V \in \mathcal{V}. Let \mathcal{G} be the following collection:

    \mathcal{G}=\left\{V \cap W: W \in \mathcal{V} \right\}

which is is open cover of V. Within the subspace V, \mathcal{G} is a point-finite open cover. By Theorem 3, \mathcal{G} must be countable. The collection \mathcal{G} represents all the open sets in \mathcal{V} that have non-empty intersection with V. Thus only countably many open sets in \mathcal{V} can meet V. So \mathcal{V} is a star-countable open refinement of \mathcal{U}. By Theorem 4, X is paracompact. \blacksquare

    Theorem 11

      Every locally compact metacompact perfectly normal space is paracompact.

Proof of Theorem 11
This follows from Theorem 10 after we prove the following two points:

  • Any locally compact space is a Baire space.
  • Any perfect locally compact space is locally CCC.

To see the first point, let Y be a locally compact space. Let W_1,W_2,W_3,\cdots be dense open sets in Y. Let y \in Y and let W \subset Y be open such that y \in W and \overline{W} is compact. We show that W contains a point that belongs to all W_n. Let X_1=W \cap W_1, which is open and non-empty. Next choose non-empty open X_2 such that \overline{X_2} \subset X_1 and X_2 \subset W_2. Next choose non-empty open X_3 such that \overline{X_3} \subset X_2 and X_3 \subset W_3. Continue in this manner, we have a sequence of open sets X_1,X_2,X_3,\cdots such that for each n, \overline{X_{n+1}} \subset X_n and \overline{X_n} is compact. The intersection of all the X_n is non-empty. The points in the intersection must belong to each W_n.

To see the second point, let Y be a locally compact space such that every closed set is a G_\delta-set. Suppose that Y is not locally CCC at y \in Y. Let U \subset Y be open such that y \in U and \overline{U} is compact. Then U must not have the CCC. Let \left\{U_\alpha: \alpha<\omega_1 \right\} be a pairwise disjoint collection of open subsets of U. Let O=\bigcup \limits_{\alpha<\omega_1} U_\alpha and let C=Y-O.

Let C=\bigcap \limits_{n=1}^\infty V_n where each V_n is open in Y and V_{n+1} \subset V_n for each integer n. For each \alpha<\omega_1, pick y_\alpha \in U_\alpha. For each y_\alpha, there is some integer f(\alpha) such that y_\alpha \notin V_{f(\alpha)}. So there must exist some integer n such that A=\left\{y_\alpha: f(\alpha)=n \right\} is uncountable.

The set A is an infinite subset of the compact set \overline{U}. So A has a limit point, say p (also called cluster point). Clearly p \notin O. So p \in C. In particular, p \in V_n. Then V_n contains some points of A. But for any y_\alpha \in A, y_\alpha \notin V_n=V_{f(\alpha)}, a contradiction. So Y must be locally CCC at each y \in Y. \blacksquare

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Some Examples

Example 1
A CCC space X with an uncountable point-finite open covers. This example demonstrates that in Theorem 2, locally-finite or locally-countable cannot be replaced by point-finite. Consider the following product space:

    Y=\prod \limits_{\alpha < \omega_1} \left\{0,1 \right\}=\left\{0,1 \right\}^{\omega_1}

i.e, the product space of \omega_1 many copies of the two-point discrete space \left\{0,1 \right\}. Let X be the set of all points h \in Y such that h(\alpha)=1 for only finitely many \alpha<\omega_1.

The product space Y is the product of separable spaces, hence has the CCC. The space X is dense in Y. Hence X has the CCC. For each \alpha<\omega_1, define U_\alpha as follows:

    U_\alpha=\left\{h \in X: h(\alpha)=1 \right\}

Then \left\{U_\alpha:\alpha<\omega_1  \right\} is a point-finite open cover of X. Of course, X in this example is not a Baire space. \blacksquare

The following three examples center around the four properties in Theorem 7 (Baire + CCC + metacompact imply Lindelof). These examples show that each property in the hypothesis is crucial.

Example 2
A separable non-Lindelof space that is a Baire space. This example shows that the metacompact assumption is crucial for Theorem 7.

The example is the Sorgenfrey plane S \times S where S is the real line with the Sorgenfrey topology (generated by the half-open intervals of the form [a,b)). It is well known that S \times S is not Lindelof. The Sorgenfrey plane is Baire and is separable (hence CCC). Furthermore, S \times S is not metacompact (if it were, it would be Lindelof by Theorem 7). \blacksquare

Example 3
A non-Lindelof metacompact Baire space M. This example shows that the CCC assumption in Theorem 7 is necessary.

This space M is the subspace of Bing’s Example G that has finite support (defined and discussed in the post A subspace of Bing’s example G. It is normal and not collectionwise normal (hence cannot be Lindelof) and metacompact. The space M does not have CCC since it has uncountably many isolated points. Any space with a dense set of isolated points is a Baire space. Thus the space M is also a Baire space. \blacksquare

Example 4
A non-Lindelof CCC metacompact non-Baire space W. This example shows that the Baire space assumption in Theorem 7 is necessary.

Let W be the set of all non-empty finite subsets of the real line with the Pixley-Roy topology. Note that W is non-Lindelof and has the CCC and is metacompact. Of course it is not Baire. For more information on Pixley-Roy spaces, see the post called Pixley-Roy hyperspaces. For the purpose of this example, the Pixley-Roy space can be built on any uncountable separable metrizable space.

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Reference

  1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  2. Tall, F. D., The Countable Chain Condition Versus Separability – Applications of Martin’s Axiom, Gen. Top. Appl., 4, 315-339, 1974.

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\copyright \ 2014 \text{ by Dan Ma}

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Pixley-Roy hyperspaces

In this post, we introduce a class of hyperspaces called Pixley-Roy spaces. This is a well-known and well studied set of topological spaces. Our goal here is not to be comprehensive but rather to present some selected basic results to give a sense of what Pixley-Roy spaces are like.

A hyperspace refers to a space in which the points are subsets of a given “ground” space. There are more than one way to define a hyperspace. Pixley-Roy spaces were first described by Carl Pixley and Prabir Roy in 1969 (see [5]). In such a space, the points are the non-empty finite subsets of a given ground space. More precisely, let X be a T_1 space (i.e. finite sets are closed). Let \mathcal{F}[X] be the set of all non-empty finite subsets of X. For each F \in \mathcal{F}[X] and for each open subset U of X with F \subset U, we define:

    [F,U]=\left\{B \in \mathcal{F}[X]: F \subset B \subset U \right\}

The sets [F,U] over all possible F and U form a base for a topology on \mathcal{F}[X]. This topology is called the Pixley-Roy topology (or Pixley-Roy hyperspace topology). The set \mathcal{F}[X] with this topology is called a Pixley-Roy space.

The hyperspace as defined above was first defined by Pixley and Roy on the real line (see [5]) and was later generalized by van Douwen (see [7]). These spaces are easy to define and is useful for constructing various kinds of counterexamples. Pixley-Roy played an important part in answering the normal Moore space conjecture. Pixley-Roy spaces have also been studied in their own right. Over the years, many authors have investigated when the Pixley-Roy spaces are metrizable, normal, collectionwise Hausdorff, CCC and homogeneous. For a small sample of such investigations, see the references listed at the end of the post. Our goal here is not to discuss the results in these references. Instead, we discuss some basic properties of Pixley-Roy to solidify the definition as well as to give a sense of what these spaces are like. Good survey articles of Pixley-Roy are [3] and [7].

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Basic Discussion

In this section, we focus on properties that are always possessed by a Pixley-Roy space given that the ground space is at least T_1. Let X be a T_1 space. We discuss the following points:

  1. The topology defined above is a legitimate one, i.e., the sets [F,U] indeed form a base for a topology on \mathcal{F}[X].
  2. \mathcal{F}[X] is a Hausdorff space.
  3. \mathcal{F}[X] is a zero-dimensional space.
  4. \mathcal{F}[X] is a completely regular space.
  5. \mathcal{F}[X] is a hereditarily metacompact space.

Let \mathcal{B}=\left\{[F,U]: F \in \mathcal{F}[X] \text{ and } U \text{ is open in } X \right\}. Note that every finite set F belongs to at least one set in \mathcal{B}, namely [F,X]. So \mathcal{B} is a cover of \mathcal{F}[X]. For A \in [F_1,U_1] \cap [F_2,U_2], we have A \in [A,U_1 \cap U_2] \subset   [F_1,U_1] \cap [F_2,U_2]. So \mathcal{B} is indeed a base for a topology on \mathcal{F}[X].

To show \mathcal{F}[X] is Hausdorff, let A and B be finite subsets of X where A \ne B. Then one of the two sets has a point that is not in the other one. Assume we have x \in A-B. Since X is T_1, we can find open sets U, V \subset X such that x \in U, x \notin V and A \cup B-\left\{ x \right\} \subset V. Then [A,U \cup V] and [B,V] are disjoint open sets containing A and B respectively.

To see that \mathcal{F}[X] is a zero-dimensional space, we show that \mathcal{B} is a base consisting of closed and open sets. To see that [F,U] is closed, let C \notin [F,U]. Either F \not \subset C or C \not \subset U. In either case, we can choose open V \subset X with C \subset V such that [C,V] \cap [F,U]=\varnothing.

The fact that \mathcal{F}[X] is completely regular follows from the fact that it is zero-dimensional.

To show that \mathcal{F}[X] is metacompact, let \mathcal{G} be an open cover of \mathcal{F}[X]. For each F \in \mathcal{F}[X], choose G_F \in \mathcal{G} such that F \in G_F and let V_F=[F,X] \cap G_F. Then \mathcal{V}=\left\{V_F: F \in \mathcal{F}[X] \right\} is a point-finite open refinement of \mathcal{G}. For each A \in \mathcal{F}[X], A can only possibly belong to V_F for the finitely many F \subset A.

A similar argument show that \mathcal{F}[X] is hereditarily metacompact. Let Y \subset \mathcal{F}[X]. Let \mathcal{H} be an open cover of Y. For each F \in Y, choose H_F \in \mathcal{H} such that F \in H_F and let W_F=([F,X] \cap Y) \cap H_F. Then \mathcal{W}=\left\{W_F: F \in Y \right\} is a point-finite open refinement of \mathcal{H}. For each A \in Y, A can only possibly belong to W_F for the finitely many F \subset A such that F \in Y.

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More Basic Results

We now discuss various basic topological properties of \mathcal{F}[X]. We first note that \mathcal{F}[X] is a discrete space if and only if the ground space X is discrete. Though we do not need to make this explicit, it makes sense to focus on non-discrete spaces X when we look at topological properties of \mathcal{F}[X]. We discuss the following points:

  1. If X is uncountable, then \mathcal{F}[X] is not separable.
  2. If X is uncountable, then every uncountable subspace of \mathcal{F}[X] is not separable.
  3. If \mathcal{F}[X] is Lindelof, then X is countable.
  4. If \mathcal{F}[X] is Baire space, then X is discrete.
  5. If \mathcal{F}[X] has the CCC, then X has the CCC.
  6. If \mathcal{F}[X] has the CCC, then X has no uncountable discrete subspaces,i.e., X has countable spread, which of course implies CCC.
  7. If \mathcal{F}[X] has the CCC, then X is hereditarily Lindelof.
  8. If \mathcal{F}[X] has the CCC, then X is hereditarily separable.
  9. If X has a countable network, then \mathcal{F}[X] has the CCC.
  10. The Pixley-Roy space of the Sorgenfrey line does not have the CCC.
  11. If X is a first countable space, then \mathcal{F}[X] is a Moore space.

Bullet points 6 to 9 refer to properties that are never possessed by Pixley-Roy spaces except in trivial cases. Bullet points 6 to 8 indicate that \mathcal{F}[X] can never be separable and Lindelof as long as the ground space X is uncountable. Note that \mathcal{F}[X] is discrete if and only if X is discrete. Bullet point 9 indicates that any non-discrete \mathcal{F}[X] can never be a Baire space. Bullet points 10 to 13 give some necessary conditions for \mathcal{F}[X] to be CCC. Bullet 14 gives a sufficient condition for \mathcal{F}[X] to have the CCC. Bullet 15 indicates that the hereditary separability and the hereditary Lindelof property are not sufficient conditions for the CCC of Pixley-Roy space (though they are necessary conditions). Bullet 16 indicates that the first countability of the ground space is a strong condition, making \mathcal{F}[X] a Moore space.

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To see bullet point 6, let X be an uncountable space. Let \left\{F_1,F_2,F_3,\cdots \right\} be any countable subset of \mathcal{F}[X]. Choose a point x \in X that is not in any F_n. Then none of the sets F_i belongs to the basic open set [\left\{x \right\} ,X]. Thus \mathcal{F}[X] can never be separable if X is uncountable.

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To see bullet point 7, let Y \subset \mathcal{F}[X] be uncountable. Let W=\cup \left\{F: F \in Y \right\}. Let \left\{F_1,F_2,F_3,\cdots \right\} be any countable subset of Y. We can choose a point x \in W that is not in any F_n. Choose some A \in Y such that x \in A. Then none of the sets F_n belongs to the open set [A ,X] \cap Y. So not only \mathcal{F}[X] is not separable, no uncountable subset of \mathcal{F}[X] is separable if X is uncountable.

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To see bullet point 8, note that \mathcal{F}[X] has no countable open cover consisting of basic open sets, assuming that X is uncountable. Consider the open collection \left\{[F_1,U_1],[F_2,U_2],[F_3,U_3],\cdots \right\}. Choose x \in X that is not in any of the sets F_n. Then \left\{ x \right\} cannot belong to [F_n,U_n] for any n. Thus \mathcal{F}[X] can never be Lindelof if X is uncountable.

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For an elementary discussion on Baire spaces, see this previous post.

To see bullet point 9, let X be a non-discrete space. To show \mathcal{F}[X] is not Baire, we produce an open subset that is of first category (i.e. the union of countably many closed nowhere dense sets). Let x \in X a limit point (i.e. an non-isolated point). We claim that the basic open set V=[\left\{ x \right\},X] is a desired open set. Note that V=\bigcup \limits_{n=1}^\infty H_n where

    H_n=\left\{F \in \mathcal{F}[X]: x \in F \text{ and } \lvert F \lvert \le n \right\}

We show that each H_n is closed and nowhere dense in the open subspace V. To see that it is closed, let A \notin H_n with x \in A. We have \lvert A \lvert>n. Then [A,X] is open and every point of [A,X] has more than n points of the space X. To see that H_n is nowhere dense in V, let [B,U] be open with [B,U] \subset V. It is clear that x \in B \subset U where U is open in the ground space X. Since the point x is not an isolated point in the space X, U contains infinitely many points of X. So choose an finite set C with at least 2 \times n points such that B \subset C \subset U. For the the open set [C,U], we have [C,U] \subset [B,U] and [C,U] contains no point of H_n. With the open set V being a union of countably many closed and nowhere dense sets in V, the open set V is not of second category. We complete the proof that \mathcal{F}[X] is not a Baire space.

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To see bullet point 10, let \mathcal{O} be an uncountable and pairwise disjoint collection of open subsets of X. For each O \in \mathcal{O}, choose a point x_O \in O. Then \left\{[\left\{ x_O \right\},O]: O \in \mathcal{O} \right\} is an uncountable and pairwise disjoint collection of open subsets of \mathcal{F}[X]. Thus if \mathcal{F}[X] is CCC then X must have the CCC.

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To see bullet point 11, let Y \subset X be uncountable such that Y as a space is discrete. This means that for each y \in Y, there exists an open O_y \subset X such that y \in O_y and O_y contains no point of Y other than y. Then \left\{[\left\{y \right\},O_y]: y \in Y \right\} is an uncountable and pairwise disjoint collection of open subsets of \mathcal{F}[X]. Thus if \mathcal{F}[X] has the CCC, then the ground space X has no uncountable discrete subspace (such a space is said to have countable spread).

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To see bullet point 12, let Y \subset X be uncountable such that Y is not Lindelof. Then there exists an open cover \mathcal{U} of Y such that no countable subcollection of \mathcal{U} can cover Y. We can assume that sets in \mathcal{U} are open subsets of X. Also by considering a subcollection of \mathcal{U} if necessary, we can assume that cardinality of \mathcal{U} is \aleph_1 or \omega_1. Now by doing a transfinite induction we can choose the following sequence of points and the following sequence of open sets:

    \left\{x_\alpha \in Y: \alpha < \omega_1 \right\}

    \left\{U_\alpha \in \mathcal{U}: \alpha < \omega_1 \right\}

such that x_\beta \ne x_\gamma if \beta \ne \gamma, x_\alpha \in U_\alpha and x_\alpha \notin \bigcup \limits_{\beta < \alpha} U_\beta for each \alpha < \omega_1. At each step \alpha, all the previously chosen open sets cannot cover Y. So we can always choose another point x_\alpha of Y and then choose an open set in \mathcal{U} that contains x_\alpha.

Then \left\{[\left\{x_\alpha \right\},U_\alpha]: \alpha < \omega_1 \right\} is a pairwise disjoint collection of open subsets of \mathcal{F}[X]. Thus if \mathcal{F}[X] has the CCC, then X must be hereditarily Lindelof.

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To see bullet point 13, let Y \subset X. Consider open sets [A,U] where A ranges over all finite subsets of Y and U ranges over all open subsets of X with A \subset U. Let \mathcal{G} be a collection of such [A,U] such that \mathcal{G} is pairwise disjoint and \mathcal{G} is maximal (i.e. by adding one more open set, the collection will no longer be pairwise disjoint). We can apply a Zorn lemma argument to obtain such a maximal collection. Let D be the following subset of Y.

    D=\bigcup \left\{A: [A,U] \in \mathcal{G} \text{ for some open } U  \right\}

We claim that the set D is dense in Y. Suppose that there is some open set W \subset X such that W \cap Y \ne \varnothing and W \cap D=\varnothing. Let y \in W \cap Y. Then [\left\{y \right\},W] \cap [A,U]=\varnothing for all [A,U] \in \mathcal{G}. So adding [\left\{y \right\},W] to \mathcal{G}, we still get a pairwise disjoint collection of open sets, contradicting that \mathcal{G} is maximal. So D is dense in Y.

If \mathcal{F}[X] has the CCC, then \mathcal{G} is countable and D is a countable dense subset of Y. Thus if \mathcal{F}[X] has the CCC, the ground space X is hereditarily separable.

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A collection \mathcal{N} of subsets of a space Y is said to be a network for the space Y if any non-empty open subset of Y is the union of elements of \mathcal{N}, equivalently, for each y \in Y and for each open U \subset Y with y \in U, there is some A \in \mathcal{N} with x \in A \subset U. Note that a network works like a base but the elements of a network do not have to be open. The concept of network and spaces with countable network are discussed in these previous posts Network Weight of Topological Spaces – I and Network Weight of Topological Spaces – II.

To see bullet point 14, let \mathcal{N} be a network for the ground space X such that \mathcal{N} is also countable. Assume that \mathcal{N} is closed under finite unions (for example, adding all the finite unions if necessary). Let \left\{[A_\alpha,U_\alpha]: \alpha < \omega_1 \right\} be a collection of basic open sets in \mathcal{F}[X]. Then for each \alpha, find B_\alpha \in \mathcal{N} such that A_\alpha \subset B_\alpha \subset U_\alpha. Since \mathcal{N} is countable, there is some B \in \mathcal{N} such that M=\left\{\alpha< \omega_1: B=B_\alpha \right\} is uncountable. It follows that for any finite E \subset M, \bigcap \limits_{\alpha \in E} [A_\alpha,U_\alpha] \ne \varnothing.

Thus if the ground space X has a countable network, then \mathcal{F}[X] has the CCC.

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The implications in bullet points 12 and 13 cannot be reversed. Hereditarily Lindelof property and hereditarily separability are not sufficient conditions for \mathcal{F}[X] to have the CCC. See [4] for a study of the CCC property of the Pixley-Roy spaces.

To see bullet point 15, let S be the Sorgenfrey line, i.e. the real line \mathbb{R} with the topology generated by the half closed intervals of the form [a,b). For each x \in S, let U_x=[x,x+1). Then \left\{[ \left\{ x \right\},U_x]: x \in S \right\} is a collection of pairwise disjoint open sets in \mathcal{F}[S].

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A Moore space is a space with a development. For the definition, see this previous post.

To see bullet point 16, for each x \in X, let \left\{B_n(x): n=1,2,3,\cdots \right\} be a decreasing local base at x. We define a development for the space \mathcal{F}[X].

For each finite F \subset X and for each n, let B_n(F)=\bigcup \limits_{x \in F} B_n(x). Clearly, the sets B_n(F) form a decreasing local base at the finite set F. For each n, let \mathcal{H}_n be the following collection:

    \mathcal{H}_n=\left\{[F,B_n(F)]: F \in \mathcal{F}[X] \right\}

We claim that \left\{\mathcal{H}_n: n=1,2,3,\cdots \right\} is a development for \mathcal{F}[X]. To this end, let V be open in \mathcal{F}[X] with F \in V. If we make n large enough, we have [F,B_n(F)] \subset V.

For each non-empty proper G \subset F, choose an integer f(G) such that [F,B_{f(G)}(F)] \subset V and F \not \subset B_{f(G)}(G). Let m be defined by:

    m=\text{max} \left\{f(G): G \ne \varnothing \text{ and } G \subset F \text{ and } G \text{ is proper} \right\}

We have F \not \subset B_{m}(G) for all non-empty proper G \subset F. Thus F \notin [G,B_m(G)] for all non-empty proper G \subset F. But in \mathcal{H}_m, the only sets that contain F are [F,B_m(F)] and [G,B_m(G)] for all non-empty proper G \subset F. So [F,B_m(F)] is the only set in \mathcal{H}_m that contains F, and clearly [F,B_m(F)] \subset V.

We have shown that for each open V in \mathcal{F}[X] with F \in V, there exists an m such that any open set in \mathcal{H}_m that contains F must be a subset of V. This shows that the \mathcal{H}_n defined above form a development for \mathcal{F}[X].

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Examples

In the original construction of Pixley and Roy, the example was \mathcal{F}[\mathbb{R}]. Based on the above discussion, \mathcal{F}[\mathbb{R}] is a non-separable CCC Moore space. Because the density (greater than \omega for not separable) and the cellularity (=\omega for CCC) do not agree, \mathcal{F}[\mathbb{R}] is not metrizable. In fact, it does not even have a dense metrizable subspace. Note that countable subspaces of \mathcal{F}[\mathbb{R}] are metrizable but are not dense. Any uncountable dense subspace of \mathcal{F}[\mathbb{R}] is not separable but has the CCC. Not only \mathcal{F}[\mathbb{R}] is not metrizable, it is not normal. The problem of finding X \subset \mathbb{R} for which \mathcal{F}[X] is normal requires extra set-theoretic axioms beyond ZFC (see [6]). In fact, Pixley-Roy spaces played a large role in the normal Moore space conjecture. Assuming some extra set theory beyond ZFC, there is a subset M \subset \mathbb{R} such that \mathcal{F}[M] is a CCC metacompact normal Moore space that is not metrizable (see Example I in [8]).

On the other hand, Pixley-Roy space of the Sorgenfrey line and the Pixley-Roy space of \omega_1 (the first uncountable ordinal with the order topology) are metrizable (see [3]).

The Sorgenfrey line and the first uncountable ordinal are classic examples of topological spaces that demonstrate that topological spaces in general are not as well behaved like metrizable spaces. Yet their Pixley-Roy spaces are nice. The real line and other separable metric spaces are nice spaces that behave well. Yet their Pixley-Roy spaces are very much unlike the ground spaces. This inverse relation between the ground space and the Pixley-Roy space was noted by van Douwen (see [3] and [7]) and is one reason that Pixley-Roy hyperspaces are a good source of counterexamples.

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Reference

  1. Bennett, H. R., Fleissner, W. G., Lutzer, D. J., Metrizability of certain Pixley-Roy spaces, Fund. Math. 110, 51-61, 1980.
  2. Daniels, P, Pixley-Roy Spaces Over Subsets of the Reals, Topology Appl. 29, 93-106, 1988.
  3. Lutzer, D. J., Pixley-Roy topology, Topology Proc. 3, 139-158, 1978.
  4. Hajnal, A., Juahasz, I., When is a Pixley-Roy Hyperspace CCC?, Topology Appl. 13, 33-41, 1982.
  5. Pixley, C., Roy, P., Uncompletable Moore spaces, Proc. Auburn Univ. Conf. Auburn, AL, 1969.
  6. Przymusinski, T., Normality and paracompactness of Pixley-Roy hyperspaces, Fund. Math. 113, 291-297, 1981.
  7. van Douwen, E. K., The Pixley-Roy topology on spaces of subsets, Set-theoretic Topology, Academic Press, New York, 111-134, 1977.
  8. Tall, F. D., Normality versus Collectionwise Normality, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 685-732, 1984.
  9. Tanaka, H, Normality and hereditary countable paracompactness of Pixley-Roy hyperspaces, Fund. Math. 126, 201-208, 1986.

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\copyright \ 2014 \text{ by Dan Ma}

A Characterization of Baire Spaces

We present a useful characterization of Baire spaces. A Baire space is a topological space X in which the conclusion of Baire category theorem holds, that is, for each countable family \left\{U_1,U_2,U_3,\cdots \right\} of open and dense subsets of X, the intersection \bigcap \limits_{n=1}^\infty U_n is dense in X. This definition is equivalent to the statement that every non-empty open subset of X is of second category in X. An elementary discussion of Baire spaces is found in this blog post. Baire spaces can also be characterized in terms of the Banach-Mazur game (see Theorem 1 in this post). We add one more characterization in terms of point-finite open cover and locally finite open family (from [2] and [3]). We prove the following theorem.

A collection \mathcal{S} of subsets of a space X is said to be point-finite if every point in the space X belongs to at most finitely many members of \mathcal{S}. A collection \mathcal{S} of subsets of X is said to be locally finite at the point x \in X if there is an open set V \subset X such that x \in V and V meets at most finitely many members of \mathcal{S}. The collection \mathcal{S} is said to be locally finite in the space X if it is locally finite at every x \in X. For any terms and concepts not explicitly defined here, refer to [1] (Engelking) or [4]) (Willard).

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

  1. X is a Baire space.
  2. For any point-finite open cover \mathcal{U} of X, the set D=\left\{x \in X: \mathcal{U} \text{ is locally finite at } x \right\} is a dense set in X.
  3. For any countable point-finite open cover \mathcal{U} of X, the set D=\left\{x \in X: \mathcal{U} \text{ is locally finite at } x \right\} is a dense set in X.

Proof
1 \Rightarrow 2
Let \mathcal{U} be a point-finite open cover of X. Let O be a non-empty open subset of X. We wish to show that O \cap D \ne \varnothing where D is the set defined in condition 2. For each n, define

\displaystyle . \ \ \ \ \ F_n=\left\{x \in O: x \text{ belongs to exactly n members of } \mathcal{U} \right\}.

Note that O=\bigcup \limits_{n=1}^\infty F_n. Since X is a Baire space, O must be of second category in X. None of the sets F_n can be a nowhere dense set. Thus for some n, F_n has non-empty interior. Choose some non-empty open set W such that W \subset F_n.

Pick y \in W. Since y \in F_n, let U_1,U_2,\cdots,U_n be the n members of \mathcal{U} that contain y. Let V=W \cap \bigcap \limits_{j=1}^n U_n. Note that V \subset W \subset F_n \subset O. Observe that V is a non-empty open set that meets exactly n members of \mathcal{U}. Therefore \mathcal{U} is locally finite at points of V, leading to the conclusion that V \subset D and O \cap D \ne \varnothing.

The direction 2 \Rightarrow 3 is immediate.

3 \Rightarrow 1
Suppose condition 3 holds. We claim that X is a Baire space. Suppose not. Let U be a non-empty open subset of X such that U=\bigcup \limits_{n=1}^\infty K_n where each K_n is nowhere dense in X. Let \mathcal{U} be defined as the following:

\displaystyle . \ \ \ \ \ \mathcal{U}=\left\{X \right\} \cup \left\{U_n: n=1,2,3,\cdots\right\},

where U_n=U - (\overline{K_1} \cup \cdots \cup \overline{K_n}). Clearly, \mathcal{U} is a point-finite open cover of X. By condition 3, D is dense in X (D is defined in condition 3). In particular, U \cap D \ne \varnothing. Choose y \in U \cap D. Since \mathcal{U} is locally finite at y, we can choose some open set V \subset U such that y \in V and such that V meets only finitely many U_j, say only up to U_1,\cdots, U_m (so V \cap U_j = \varnothing for all j > m).

On the other hand, all sets K_j are nowhere dense. So we can choose some open set V_0 \subset V such that V_0 misses the nowhere dense set \overline{K_1} \cup \cdots \cup \overline{K_m} \cup \overline{K_{m+1}}. In particular, this means that V_0 \cap U_{m+1} \ne \varnothing, contradicting that V \cap U_j = \varnothing for all j > m. So X must be a Baire space if condition 3 holds. \blacksquare

Reference

  1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  2. Fletcher, P., Lindgren, W. F., A note on spaces of second category, Arch, Math., 24, 186-187, 1973.
  3. McCoy, R. A., A Baire space extension, Proc. Amer. Math. Soc., 33, 199-202, 1972.
  4. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

Baire Category Theorem and the Finite Intersection Property

A Baire space is a topological space in which the intersection of any countable family of open and dense sets is dense (equivalently every non-empty open subset is of second category). One version of the Baire category theorem states that every complete metric space is a Baire space. Another common version states that every compact Hausdorff space is a Baire space. Another version states that every locally compact Hausdorff space is a Baire space. The commonality among these versions is the finite intersection property (whenever a collection of a certain type of sets satisfies the property that any finite subcollection has non-empty intersection, the whole collection has non-empty intersection). For each of these classes of spaces, in addition to countably compact spaces and pseudocompact spaces, Baire category theorem is derived from having one specific form of the finite intersection property. In this post, we explore this relationship.

In each of the following theorem pairs, the B Theorem follows from the A theorem. The A theorem is a form of the finite intersection property and the B theorem is a version of Baire category theorem.

Another interesting observation is that the finite intersection properties discussed here can give a stronger property than being a Baire space. This stronger property is defined by the Banach-Mazur game.

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Theorem 1A
Let (X, \rho) be a metric space. Then the following conditions are equivalent.

  1. (X, \rho) is a complete metric space.
  2. For each decreasing sequence C_1 \supset C_2 \supset C_3 \supset \cdots of non-empty closed subsets of X such that the diameters of the sets C_n converge to zero, we have \bigcap \limits_{n=1}^\infty C_n \ne \varnothing.

Theorem 1B
Every complete metric space is a Baire space.

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Theorem 2A
Let X be a Hausdorff space. Then the following conditions are equivalent.

  1. X is a compact space.
  2. For every family \mathcal{F} consisting of non-empty closed subsets of X, if \mathcal{F} has the finite intersection property, then \mathcal{F} has non-empty intersection.

Theorem 2B

  • Every compact Hausdorff space is a Baire space.
  • Every locally compact Hausdorff space is a Baire space.

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Theorem 3A
Let X be a Hausdorff space. Then the following conditions are equivalent.

  1. X is a countably compact space.
  2. For every countable family \mathcal{F} consisting of non-empty closed subsets of X, if \mathcal{F} has the finite intersection property, then \mathcal{F} has non-empty intersection.
  3. For each decreasing sequence C_1 \supset C_2 \supset C_3 \supset \cdots of non-empty closed subsets of X, we have \bigcap \limits_{n=1}^\infty C_n \ne \varnothing.

Theorem 3B
Every countably compact Hausdorff space is a Baire space.

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Theorem 4A
Let X be a regular 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.

Theorem 4B
Every regular pseudocompact space is a Baire space.

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Remark
Theorem 1A (the Cantor Theorem) can be found in Engelking (page 269 in [1]). Theorem 2A and Theorem 3A can also be found in Engelking (they are also proved in this post). Theorem 4B is also found in Engelking (Theorem 3.10.23 in page 207 of [1]) and is proved this post.

We would like to explicitly point out that between Thoerem 1A and Theorem 2A, none of the two theorems implies the other. For example, even though both complete metric spaces and compact Hausdorff spaces are Baire spaces, complete metric spaces are not necessarily compact and there are compact spaces that are not even metrizable. However, the finite intersection property of Theorem 2A implies that of Theorem 3A, which in turn implies the finite intersection property of Theorem 4A.
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Baire Category Theorem

The proofs of all four B theorems are amazingly similar. It is a matter of exploiting the fact that whenever a decreasing sequence of open sets satisfying the condition that each closure is a subset of the previous open set (and satisfying some other condition), the sequence of open sets has non-empty intersection. For example, for complete metric space, make sure that the closures of the open sets have diameters going to zero. For any reader who is new to this material, it will be very instructive to walk through the arguments of these Baire category theorems. The proof of Theorem 1A can be found this post. We prove Theorem 4B.

Recall that X is a Baire space if \left\{U_1,U_2,U_3,\cdots \right\} is a countable family of open and dense sets in X, \bigcap \limits_{i=1}^\infty U_i is dense in X, or equivalently every non-empty open subset of X is of second category in X. For more background about the concepts of Baire space and category (see [1] or this post).

Proof of Theorem 4B
Let X be a regular pseudocompact space. Let \left\{U_1,U_2,U_3,\cdots \right\} be a countable family of open and dense sets in X. Let O be a non-empty open subset of X. We show that O has to contain points of \bigcap \limits_{n=1}^\infty U_n. We let O_1=O \cap U_1. We find open O_2 such that O_2 \subset U_2 and \overline{O_2} \subset O_1 (using regularity). Continue this inductive process, we have for each n, an open O_n such that O_n \subset U_n and \overline{O_n} \subset O_{n-1}. Then we have a decreasing sequence of open sets O_n as in condition 2 of Theorem 4A. Then we have \bigcap \limits_{n=1}^\infty \overline{O_n} \ne \varnothing. Since \overline{O_{n+1}} \subset O_n for each n, we also have \bigcap \limits_{n=1}^\infty O_n \ne \varnothing. It is clear that \bigcap \limits_{n=1}^\infty O_n \subset \bigcap \limits_{n=1}^\infty U_n. \blacksquare

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Banach-Mazur Game

In proving the above versions of Baire category theorem, we can exploit the appropriate version of the finite intersection property – the situation that any nested decreasing sequence of open sets (under some specified conditions) has non-empty intersection. In fact, the finite intersection property offers more than just Baire category theorem; it can endow the space in question a type of completeness property stronger than Baire space. This completeness property is defined using 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. For more detailed discussion of the game, see this post.

One interesting point that we like to make about the finite intersection property ranging from Theorem 1A to Theorem 4A is that the player \alpha can always win the Banach-Mazur game as long as he/she plays the game according to each specific version of the finite intersection. For example, playing the game in a complete metric space, player \alpha always wins as long as he/she makes the diameters of the closures of the open sets going to zero. In a regular pseudocompact space, player \alpha can always win by making the closure of each of his/her open sets a subset of the previous move of other player.

A topological space in which the player \alpha has a winning strategy is said to be a weakly \alpha-favorable space. Thus complete metric spaces, compact Hausdorff spaces, locally compact Hausdorff spaces, countably compact Hausdorff spaces, regular pseudocompact spaces are all weakly \alpha-favorable.

There is characterization of Baire spaces in terms of the Banach-Mazur game. A space X is a Baire space if and only if the player \beta has no winning strategy in the Banach-Mazur game played on the space X (see theorem 1 in this post). If the player \alpha can always win, then player \beta can never win. In terms of game terminology, if player \alpha has a winning strategy, then the other player (player \beta) has no winning strategy. Thus a space is weakly \alpha-favorable implies that it is a Baire space. But the implication is not reversible (see example in this post).

So all the spaces discussed from Theorem 1A to Theorem 4A are all weakly \alpha-favorable, a property stronger than Baire spaces. These observations are summarized in the following theorems.

Theorem 1C
Every complete metric space is a weakly \alpha-favorable space.

Theorem 2C

  • Every compact Hausdorff space is a weakly \alpha-favorable space.
  • Every locally compact Hausdorff space is a weakly \alpha-favorable space.

Theorem 3C
Every countably compact Hausdorff space is a weakly \alpha-favorable space.

Theorem 4C
Every regular pseudocompact space is a weakly \alpha-favorable space.

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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.

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<c.

Let B=\left\{ x_\alpha: \alpha<c \right\}. 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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Conclusions about Bernstein Sets

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.

A Remark About Topological Games

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}

A Question About The Rational Numbers

Let \mathbb{R} be the real line and \mathbb{Q} be the set of all rational numbers. Consider the following question:

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Question

  • For each nonnegative integer n, let U_n be an open subset of \mathbb{R} such that that \mathbb{Q} \subset U_n. The intersection \bigcap \limits_{n=0}^\infty U_n is certainly nonempty since it contains \mathbb{Q}. Does this intersection necessarily contain some irrational numbers?

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While taking a real analysis course, the above question was posted to the author of this blog by the professor. Indeed, the question is an excellent opening of the subject of category. We first discuss the Baire category theorem and then discuss the above question. A discussion of Baire spaces follow. For any notions not defined here and for detailed discussion of any terms discussed here, see [1] and [2].

In the above question, the set \bigcap \limits_{n=0}^\infty U_n is a G_\delta set since it is the intersection of countably many open sets. It is also dense in the real line \mathbb{R} since it contains the rational numbers. So the question can be rephrased as: is the set of rational numbers \mathbb{Q} a G_\delta set? Can a dense G_\delta set in the real line \mathbb{R} be a “small” set such as \mathbb{Q}? The discussion below shows that \mathbb{Q} is too “thin” to be a dense G_\delta set. Put it another way, a dense G_\delta subset of the real line is a “thick” set. First we present the Baire category theorem.
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Baire Category Theorem

Let X be a complete metric space. For each nonnegative integer n, let O_n be an open subset of X that is also dense in X. Then \bigcap \limits_{n=0}^\infty O_n is dense in X.

Proof
Let A=\bigcap \limits_{n=0}^\infty O_n. Let V_0 be any nonempty open subset of X. We show that V_0 contains some point of A.

Since O_0 is dense in X, V_0 contains some point of O_0. Let x_0 be one such point and choose open set V_1 such that x_0 \in V_1 and \overline{V_1} \subset V_0 \cap O_0 \subset V_0 with the additional condition that the diameter of \overline{V_1} is less than \displaystyle \frac{1}{2^1}.

Since O_1 is dense in X, V_1 contains some point of O_1. Let x_1 be one such point and choose open set V_2 such that x_1 \in V_2 and \overline{V_2} \subset V_1 \cap O_1 \subset V_1 with the additional condition that the diameter of \overline{V_2} is less than \displaystyle \frac{1}{2^2}.

By continuing this inductive process, we obtain a nested sequence of open sets V_n and a sequence of points x_n such that x_n \in V_n \subset \overline{V_n} \subset V_{n-1} \cap O_{n-1} \subset V_0 for each n and that the diameters of \overline{V_n} converge to zero (according to some complete metric on X). Then the sequence of points x_n is a Cauchy sequence. Since X is a complete metric space, the sequence x_n converges to a point x \in X.

We claim that x \in V_0 \cap A. To see this, note that for each n, x_j \in \overline{V_n} for each j \ge n. Since x is the sequential limit of x_j, x \in \overline{V_n} for each n. It follows that x \in O_n for each n (x \in A) and x \in V_0. This completes the proof of Baire category theorem. \blacksquare

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Discussion of the Above Question

For each nonnegative integer n, let U_n be an open subset of \mathbb{R} such that that \mathbb{Q} \subset U_n. We claim that the intersection \bigcap \limits_{n=0}^\infty U_n contain some irrational numbers.

Suppose the intersection contains no irrational numbers, that is, \mathbb{Q}=\bigcap \limits_{n=0}^\infty U_n.

Let \mathbb{Q} be enumerated by \left\{r_0,r_1,r_2,\cdots \right\}. For each n, let G_n=\mathbb{R}-\left\{ r_n \right\}. Then each G_n is an open and dense set in \mathbb{R}. Note that the set of irrational numbers \mathbb{P}=\bigcap \limits_{n=0}^\infty G_n.

We then have countably many open and dense sets U_0,U_1,U_2,\cdots,G_0,G_1,G_2,\cdots whose intersection is empty. Note that any point that belongs to all U_n has to be a rational number and any point that belongs to all G_n has to be an irrational number. On the other hand, the real line \mathbb{R} with the usual metric is a complete metric space. By the Baire category theorem, the intersection of all U_n and G_n must be nonempty. Thus the intersection \bigcap \limits_{n=0}^\infty U_n must contain more than rational numbers.

It follows that the set of rational numbers \mathbb{Q} cannot be a G_\delta set in \mathbb{R}. In fact, the discussion below will show that the in a complete metric space such as the real line, any dense G_\delta set must be a “thick” set (see Theorem 3 below).
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Baire Spaces

The version of the Baire category theorem discussed above involves complete metric spaces. However, the ideas behind the Baire category theorem are topological in nature. The following is the conclusion of the Baire category theorem:

(*) \ \ \ \ X is a topological space such that for each countable family \left\{U_0,U_1,U_2,\cdots \right\} of open and dense sets in X, the intersection \bigcap \limits_{n=0}^\infty U_n is dense in X.

A Baire space is a topological space in which the condition (*) holds. The Baire category theorem as stated above gives a sufficient condition for a topological space to be a Baire space. There are plenty of Baire spaces that are not complete metric spaces, in fact, not even metric spaces. The condition (*) is a topological property. In order to delve deeper into this property, let’s look at some related notions.

Let X be a topological space. A set A \subset X is dense in X if every open subset of X contains a point of A (i.e. \overline{A}=X). A set A \subset X is nowhere dense in X if for every open subset U of X, there is some open set V \subset U such that V contains no point of A (another way to describe this: \overline{A} contains no interior point of X).

A set is dense if its points can be found in every nonempty open set. A set is nowhere dense if every nonempty open set has an open subset that misses it. For example, the set of integers \mathbb{N} is nowhere dense in \mathbb{R}.

A set A \subset X is of first category in X if A is the union of countably many nowhere dense sets in X. A set A \subset X is of second category in X if it is not of first category in X.

To make sense of these notions, the following observation is key:

(**) \ \ \ \ F \subset X is nowhere dense in X if and only if X-\overline{F} is an open and dense set in X.

So in a Baire space, if you take away any countably many closed and nowhere dense sets (in other words, taking away a set of first category in X), there is a remainder (there are still points remaining) and the remainder is still dense in X. In thinking of sets of first category as “thin”, a Baire space is one that is considered “thick” or “fat” in that taking away a “thin” set still leaves a dense set.

A space X is of second category in X means that if you take away any countably many closed and nowhere dense sets in X, there are always points remaining. For a set Y \subset X, Y is of second category in X means that if you take away from Y any countably many closed and nowhere dense sets in X, there are still points remaining in Y. A set of second category is “thick” in the sense that after taking away a “thin” set there are still points remaining.

For example, \mathbb{N} is nowhere dense in \mathbb{R} and thus of first category in \mathbb{R}. However, \mathbb{N} is of second category in itself. In fact, \mathbb{N} is a Baire space since it is a complete metric space (with the usual metric).

For example, \mathbb{Q} is of first category in \mathbb{R} since it is the union of countably many singleton sets (\mathbb{Q} is also of first category in itself).

For example, let T=[0,1] \cup (\mathbb{Q} \cap [2,3]). The space T is not a Baire space since after taking away the rational numbers in [2,3], the remainder is no longer dense in T. However, T is of second category in itself.

For example, any Cantor set defined in the real line is nowhere dense in \mathbb{R}. However, any Cantor set is of second category in itself (in fact a Baire space).

The following theorems summarize these concepts.

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

  1. X is of second category in itself.
  2. The intersection of countably many dense open sets is nonempty.

Theorem 1b
Let X be a topological space. Let A \subset X. The following conditions are equivalent:

  1. The set A is of second category in X.
  2. The intersection of countably many dense open sets in X must intersect A.

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

  1. X is a Baire space, i.e., the intersection of countably many dense open sets is dense in X.
  2. Every nonempty open subset of X is of second category in X.

The above theorems can be verified by appealing to the relevant definitions, especially the observation (**). Theorems 2 and 1a indicate that any Baire space is of second category in itself. The converse is not true (see the space T=[0,1] \cup (\mathbb{Q} \cap [2,3]) discussed above).

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Dense G delta Subsets of a Baire Space

In answering the question stated at the beginning, we have shown that \mathbb{Q} cannot be a G_\delta set. Being a set of first category, \mathbb{Q} cannot be a dense G_\delta set. In fact, it can be shown that in a Baire space, any dense G_\delta subset is also a Baire space.

Theorem 3
Let X be a Baire space. Then any dense G_\delta subset of X is also a Baire space.

Proof
Let Y=\bigcap \limits_{n=0}^\infty U_n where each U_n is open and dense in X. We show that Y is a Baire space. In light of Theorem 2, we show that every nonempty open set of Y is of second category in Y.

Soppuse that there is a nonempty open subset U \subset Y such that U is of first category in Y. Then U=\bigcup \limits_{n=0}^\infty W_n where each W_n is nowhere dense in Y. It can be shown that each W_n is also nowhere dense in X.

Since U is open in Y, there is an open set U^* \subset X such that U^* \cap Y=U. Note that for each n, F_n=X-U_n is closed and nowhere dense in X. Then we have:

\displaystyle (1) \ \ \ \ \ U^*=\bigcup \limits_{n=0}^\infty (F_n \cap U^*) \cup \bigcup \limits_{n=0}^\infty W_n

(1) shows that U^* is the union of countably many nowhere dense sets in X, contracting that every nonempty subset of X is of second category in X. Thus we can conclude that every nonempty open subset of Y is of second category in Y. \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.