Compact metrizable scattered spaces

A scattered space is one in which there are isolated points found in every subspace. Specifically, a space X is a scattered space if every non-empty subspace Y of X has a point y \in Y such that y is an isolated point in Y, i.e. the singleton set \left\{y \right\} is open in the subspace Y. A handy example is a space consisting of ordinals. Note that in a space of ordinals, every non-empty subset has an isolated point (e.g. its least element). In this post, we discuss scattered spaces that are compact metrizable spaces.

Here’s what led the author to think of such spaces. Consider Theorem III.1.2 found on page 91 of Arhangelskii’s book on topological function space [1], which is Theorem 1 stated below:

Thereom 1
For any compact space X, the following conditions are equivalent:

  • The function space C_p(X) is a Frechet-Urysohn space.
  • The function space C_p(X) is a k space.
  • X is a scattered space.

Let’s put aside the Frechet-Urysohn property and the k space property for the moment. For any Hausdorff space X, let C(X) be the set of all continuous real-valued functions defined on the space X. Since C(X) is a subspace of the product space \mathbb{R}^X, a natural topology that can be given to C(X) is the subspace topology inherited from the product space \mathbb{R}^X. Then C_p(X) is simply the set C(X) with the product subspace topology (also called the pointwise convergence topology).

Let’s say the compact space X is countable and infinite. Then the function space C_p(X) is metrizable since it is a subspace of \mathbb{R}^X, a product of countably many lines. Thus the function space C_p(X) has the Frechet-Urysohn property (being metrizable implies Frechet-Urysohn). This means that the compact space X is scattered. The observation just made is a proof that any infinite compact space that is countable in cardinality must be scattered. In particular, every infinite compact and countable space must have an isolated point. There must be a more direct proof of this same fact without taking the route of a function space. The indirect argument does not reveal the essential nature of compact metric spaces. The essential fact is that any uncountable compact metrizable space contains a Cantor set, which is as unscattered as any space can be. Thus the only scattered compact metrizable spaces are the countable ones.

The main part of the proof is the construction of a Cantor set in a compact metrizable space (Theorem 3). The main result is Theorem 4. In many settings, the construction of a Cantor set is done in the real number line (e.g. the middle third Cantor set). The construction here is in a more general setting. But the idea is still the same binary division process – the splitting of a small open set with compact closure into two open sets with disjoint compact closure. We also use that fact that any compact metric space is hereditarily Lindelof (Theorem 2).

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Compact metrizable spaces

We first define some notions before looking at compact metrizable spaces in more details. Let X be a space. Let A \subset X. Let p \in X. We say that p is a limit point of A if every open subset of X containing p contains a point of A distinct from p. So the notion of limit point here is from a topology perspective and not from a metric perspective. In a topological space, a limit point does not necessarily mean that it is the limit of a convergent sequence (however, it does in a metric space). The proof of the following theorem is straightforward.

Theorem 2
Let X be a hereditarily Lindelof space (i.e. every subspace of X is Lindelof). Then for any uncountable subset A of X, all but countably many points of A are limit points of A.

We now discuss the main result.

Theorem 3
Let X be a compact metrizable space such that every point of X is a limit point of X. Then there exists an uncountable closed subset C of X such that every point of C is a limit point of C.

Proof of Theorem 3
Note that any compact metrizable space is a complete metric space. Consider a complete metric \rho on the space X. One fact that we will use is that if there is a sequence of closed sets X \supset H_1 \supset H_2 \supset H_3 \supset \cdots such that the diameters of the sets H (based on the complete metric \rho) decrease to zero, then the sets H_n collapse to one point.

The uncountable closed set C we wish to define is a Cantor set, which is constructed from a binary division process. To start, pick two points p_0,p_1 \in X such that p_0 \ne p_1. By assumption, both points are limit points of the space X. Choose open sets U_0,U_1 \subset X such that

  • p_0 \in U_0,
  • p_1 \in U_1,
  • K_0=\overline{U_0} and K_1=\overline{U_1},
  • K_0 \cap K_1 = \varnothing,
  • the diameters for K_0 and K_1 with respect to \rho are less than 0.5.

Note that each of these open sets contains infinitely many points of X. Then we can pick two points in each of U_0 and U_1 in the same manner. Before continuing, we set some notation. If \sigma is an ordered string of 0’s and 1’s of length n (e.g. 01101 is a string of length 5), then we can always extend it by tagging on a 0 and a 1. Thus \sigma is extended as \sigma 0 and \sigma 1 (e.g. 01101 is extended by 011010 and 011011).

Suppose that the construction at the nth stage where n \ge 1 is completed. This means that the points p_\sigma and the open sets U_\sigma have been chosen such that p_\sigma \in U_\sigma for each length n string of 0’s and 1’s \sigma. Now we continue the picking for the (n+1)st stage. For each \sigma, an n-length string of 0’s and 1’s, choose two points p_{\sigma 0} and p_{\sigma 1} and choose two open sets U_{\sigma 0} and U_{\sigma 1} such that

  • p_{\sigma 0} \in U_{\sigma 0},
  • p_{\sigma 1} \in U_{\sigma 1},
  • K_{\sigma 0}=\overline{U_{\sigma 0}} \subset U_{\sigma} and K_{\sigma 1}=\overline{U_{\sigma 1}} \subset U_{\sigma},
  • K_{\sigma 0} \cap K_{\sigma 1} = \varnothing,
  • the diameters for K_{\sigma 0} and K_{\sigma 1} with respect to \rho are less than 0.5^{n+1}.

For each positive integer m, let C_m be the union of all K_\sigma over all \sigma that are m-length strings of 0’s and 1’s. Each C_m is a union of finitely many compact sets and is thus compact. Furthermore, C_1 \supset C_2 \supset C_3 \supset \cdots. Thus C=\bigcap \limits_{m=1}^\infty C_m is non-empty. To complete the proof, we need to show that

  • C is uncountable (in fact of cardinality continuum),
  • every point of C is a limit point of C.

To show the first point, we define a one-to-one function f: \left\{0,1 \right\}^N \rightarrow C where N=\left\{1,2,3,\cdots \right\}. Note that each element of \left\{0,1 \right\}^N is a countably infinite string of 0’s and 1’s. For each \tau \in \left\{0,1 \right\}^N, let \tau \upharpoonright  n denote the string of the first n digits of \tau. For each \tau \in \left\{0,1 \right\}^N, let f(\tau) be the unique point in the following intersection:

    \displaystyle \bigcap \limits_{n=1}^\infty K_{\tau \upharpoonright  n} = \left\{f(\tau) \right\}

This mapping is uniquely defined. Simply conceptually trace through the induction steps. For example, if \tau are 01011010…., then consider K_0 \supset K_{01} \supset K_{010} \supset \cdots. At each next step, always pick the K_{\tau \upharpoonright  n} that matches the next digit of \tau. Since the sets K_{\tau \upharpoonright  n} are chosen to have diameters decreasing to zero, the intersection must have a unique element. This is because we are working in a complete metric space.

It is clear that the map f is one-to-one. If \tau and \gamma are two different strings of 0’s and 1’s, then they must differ at some coordinate, then from the way the induction is done, the strings would lead to two different points. It is also clear to see that the map f is reversible. Pick any point x \in C. Then the point x must belong to a nested sequence of sets K‘s. This maps to a unique infinite string of 0’s and 1’s. Thus the set C has the same cardinality as the set \left\{0,1 \right\}^N, which has cardinality continuum.

To see the second point, pick x \in C. Suppose x=f(\tau) where \tau \in \left\{0,1 \right\}^N. Consider the open sets U_{\tau \upharpoonright n} for all positive integers n. Note that x \in U_{\tau \upharpoonright n} for each n. Based on the induction process described earlier, observe these two facts. This sequence of open sets has diameters decreasing to zero. Each open set U_{\tau \upharpoonright n} contains infinitely many other points of C (this is because of all the open sets U_{\tau \upharpoonright k} that are subsets of U_{\tau \upharpoonright n} where k \ge n). Because the diameters are decreasing to zero, the sequence of U_{\tau \upharpoonright n} is a local base at the point x. Thus, the point x is a limit point of C. This completes the proof. \blacksquare

Theorem 4
Let X be a compact metrizable space. It follows that X is scattered if and only if X is countable.

Proof of Theorem 4
\Longleftarrow
In this direction, we show that if X is countable, then X is scattered (the fact that can be shown using the function space argument pointed out earlier). Here, we show the contrapositive: if X is not scattered, then X is uncountable. Suppose X is not scattered. Then every point of X is a limit point of X. By Theorem 3, X would contain a Cantor set C of cardinality continuum.

\Longrightarrow
In this direction, we show that if X is scattered, then X is countable. We also show the contrapositive: if X is uncountable, then X is not scattered. Suppose X is uncountable. By Theorem 2, all but countably many points of X are limit points of X. After discarding these countably many isolated points, we still have a compact space. So we can just assume that every point of X is a limit point of X. Then by Theorem 3, X contains an uncountable closed set C such that every point of C is a limit point of C. This means that X is not scattered. \blacksquare

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Remarks

A corollary to the above discussion is that the cardinality for any compact metrizable space is either countable (including finite) or continuum (the cardinality of the real line). There is nothing in between or higher than continuum. To see this, the cardinality of any Lindelof first countable space is at most continuum according to a theorem in this previous post (any compact metric space is one such). So continuum is an upper bound on the cardinality of compact metric spaces. Theorem 3 above implies that any uncountable compact metrizable space has to contain a Cantor set, hence has cardinality continuum. So the cardinality of a compact metrizable space can be one of two possibilities – countable or continuum. Even under the assumption of the negation of the continuum hypothesis, there will be no uncountable compact metric space of cardinality less than continuum. On the other hand, there is only one possibility for the cardinality of a scattered compact metrizable, which is countable.

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Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.

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\copyright \ 2015 \text{ by Dan Ma}

Cartesian Products of Two Paracompact Spaces – Continued

Consider the real line \mathbb{R} with a topology finer than the usual topology obtained by isolating each point in \mathbb{P} where \mathbb{P} is the set of all irrational numbers. The real line with this finer topology is called the Michael line and we use \mathbb{M} to denote this topological space. It is a classic result that \mathbb{M} \times \mathbb{P} is not normal (see “Michael Line Basics”). Even though the Michael line \mathbb{M} is paracompact (it is in fact hereditarily paracompact), \mathbb{M} is not perfectly normal. Result 3 below will imply that the Michael line cannot be perfectly normal. Otherwise \mathbb{M} \times \mathbb{P} would be paracompact (hence normal). Result 3 is the statement that if X is paracompact and perfectly normal and Y is a metric space then X \times Y is paracompact and perfectly normal. We also use this result to show that if X is hereditarily Lindelof and Y is a separable metric space, then X \times Y is hereditarily Lindelof (see Result 4 below).

This post is a continuation of the post “Cartesian Products of Two Paracompact Spaces”. In that post, four results are listed. They are:

Result 1

    If X is paracompact and Y is compact, then X \times Y is paracompact.

Result 2

    If X is paracompact and Y is \sigma-compact, then X \times Y is paracompact.

Result 3

    If X is paracompact and perfectly normal and Y is metrizable, then X \times Y is paracompact and perfectly normal.

Result 4

    If X is hereditarily Lindelof and Y is a separable metric space, then X \times Y is hereditarily Lindelof.

Result 1 and Result 2 are proved in the previous post “Cartesian Products of Two Paracompact Spaces”. Result 3 and Result 4 are proved in this post. All spaces are assumed to be regular.

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Paracompact Spaces, Lindelof Spaces and Other Information

A paracompact space is one in which every open cover has a locally finite open refinement. The previous post “Cartesian Products of Two Paracompact Spaces” has a basic discussion on paracompact spaces. For the sake of completeness, we repeat here some of the results discussed in that post. A proof of Proposition 1 can be found in [1] (Theorem 5.1.11 in page 302) or in [2] (Theorem 20.7 in page 146).. For a proof of Proposition 2, see Theorem 3 in the previous post “Cartesian Products of Two Paracompact Spaces”. We provide a proof for Proposition 3.

Proposition 1
Let X be a regular space. Then X is paracompact if and only if every open cover \mathcal{U} of X has a \sigma-locally finite open refinement.

Proposition 2
Every F_\sigma-subset of a paracompact space is paracompact.

Proposition 3
Any paracompact space with a dense Lindelof subspace is Lindelof.

Proof of Proposition 3
Let L be a paracompact space. Let M \subset L be a dense Lindelof subspace. Let \mathcal{U} be an open cover of L. Since we are working with a regular space, let \mathcal{V} be an open cover of L such that \left\{\overline{V}: V \in \mathcal{V} \right\} refines \mathcal{U}. Let \mathcal{W} be a locally finite open refinement of \mathcal{V}. Choose \left\{W_1,W_2,W_3,\cdots \right\} \subset \mathcal{W} such that it is a cover of M. Since M \subset \bigcup \limits_{i=1}^\infty W_i, \overline{\bigcup \limits_{i=1}^\infty W_i}=L.

Since the sets W_i come from a locally finite collection, they are closure preserving. Hence we have:

    \overline{\bigcup \limits_{i=1}^\infty W_i}=\bigcup \limits_{i=1}^\infty \overline{W_i}=L

For each i, choose some U_i \in \mathcal{U} such that \overline{W_i} \subset U_i. Then \left\{U_1,U_2,U_3,\cdots \right\} is a countable subcollection of \mathcal{U} covering the space L. \blacksquare

A space is said to be a perfectly normal if it is a normal space with the additional property that every closed subset is a G_\delta-set in the space (equivalently every open subset is an F_\sigma-set). We need two basic results about hereditarily Lindelof spaces. A space is Lindelof if every open cover of that space has a countable subcover. A space is hereditarily Lindelof if every subspace of that space is Lindelof. Proposition 4 below, stated without proof, shows that to prove a space is hereditarily Lindelof, we only need to show that every open subspace is Lindelof.

Proposition 4
Let L be a space. Then L is hereditarily Lindelof if and only if every open subspace of L is Lindelof.

Proposition 5
Let L be a Lindelof space. Then L is hereditarily Lindelof if and only if L is perfectly normal.

Proof of Proposition 5
\Rightarrow Suppose L is hereditarily Lindelof. It is well known that regular Lindelof space is normal. Thus L is normal. It remains to show that every open subset of L is F_\sigma. Let U \subset L be an non-empty open set. For each x \in U, let V_x be open such that x \in V_x and \overline{V_x} \subset U (the space is assumed to be regular). By assumption, the open set U is Lindelof. The open sets V_x form an open cover of U. Thus U is the union of countably many \overline{V}_x.

\Leftarrow Suppose L is perfectly normal. To show that L is hereditarily Lindelof, it suffices to show that every open subset of L is Lindelof (by Proposition 4). Let U \subset L be non-empty open. By assumption, U=\bigcup \limits_{i=1}^\infty F_i where each F_i is a closed set in L. Since the Lindelof property is hereditary with respect to closed subsets, U is Lindelof. \blacksquare

Another important piece of information that we need is the following metrization theorem. It shows that being a metrizable space is equivalent to have a base that is \sigma-locally finite. In proving Result 3, we will assume that the metric factor has such a base. This is a classic metrization theorem (see [1] or [2] or any other standard topology text).

Theorem 6
Let X be a space. Then X is metrizable if and only if X has a \sigma-locally finite base.

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Result 3

Result 3 is the statement that:

    If X is paracompact and perfectly normal and Y is a metric space then X \times Y is paracompact and perfectly normal.

Result 3 follows from the following two lemmas.

Lemma 7
If the following two conditions hold:

  • every open subset of X is an F_\sigma-set in X,
  • Y is a metric space,

then every open subset of X \times Y is an F_\sigma-set in X \times Y.

Proof of Lemma 7
Let U be a open subset of X \times Y. If U=\varnothing, then U is certainly the union of countably many closed sets. So assume U \ne \varnothing. Let \mathcal{B}=\bigcup \limits_{i=1}^\infty \mathcal{B}_i be a base for Y such that each \mathcal{B}_i is locally finite in Y (by Theorem 6, such a base exists since Y is metrizable).

Consider all non-empty B \in \mathcal{B} such that we can choose nonempty open set W_B \subset X with W_B \times \overline{B} \subset U. Since U is non-empty open, such pairs (B, W_B) exist. Let \mathcal{B}^* be the collection of all non-empty B \in \mathcal{B} for which there is a matching non-empty W_B. For each i, let \mathcal{B}_i^*=\mathcal{B}^* \cap \mathcal{B}_i. Of course, each \mathcal{B}_i^* is still locally finite.

Since every open subset of X is an F_\sigma-set in X, for each W_B, we can write W_B as

    W_B=\bigcup \limits_{j=1}^\infty W_{B,j}

where each W_{B,i} is closed in X.

For each i=1,2,3,\cdots and each j=1,2,3,\cdots, consider the following collection:

    \mathcal{V}_{i,j}=\left\{W_{B,j} \times \overline{B}: B \in \mathcal{B}_i^* \right\}

Each element of \mathcal{V}_{i,j} is a closed set in X \times Y. Since \mathcal{B}_i^* is a locally finite collection in Y, \mathcal{V}_{i,j} is a locally finite collection in X \times Y. Define V_{i,j}=\bigcup \mathcal{V}_{i,j}. The set V_{i,j} is a union of closed sets. In general, the union of closed sets needs not be closed. However, V_{i,j} is still a closed set in X \times Y since \mathcal{V}_{i,j} is a locally finite collection of closed sets. This is because a locally finite collection of sets is closure preserving. Note the following:

    \overline{V_{i,j}}=\overline{\bigcup \mathcal{V}_{i,j}}=\overline{\bigcup \left\{W_{B,j} \times \overline{B}: B \in \mathcal{B}_i^* \right\}}=\bigcup \left\{\overline{W_{B,j} \times \overline{B}}: B \in \mathcal{B}_i^* \right\}

      =\bigcup \left\{W_{B,j} \times \overline{B}: B \in \mathcal{B}_i^* \right\}=V_{i,j}

Finally, we have U=\bigcup \limits_{i=1}^\infty \bigcup \limits_{j=1}^\infty V_{i,j}, which is the union of countably many closed sets. \blacksquare

Lemma 8
If X is a paracompact space satisfying the following two conditions:

  • every open subset of X is an F_\sigma-set in X,
  • Y is a metric space,

then X \times Y is paracompact.

Proof of Lemma 8
As in the proof of the above lemma, let \mathcal{B}=\bigcup \limits_{i=1}^\infty \mathcal{B}_i be a base for Y such that each \mathcal{B}_i is locally finite in Y. Let \mathcal{U} be an open cover of X \times Y. Assume that elements of \mathcal{U} are of the form A \times B where A is open in X and B \in \mathcal{B}.

For each B \in \mathcal{B}, consider the following two items:

    \mathcal{W}_B=\left\{A: A \times B \in \mathcal{U} \right\}

    W_B=\bigcup \mathcal{W}_B

To simplify matter, we only consider B \in \mathcal{B} such that \mathcal{W}_B \ne \varnothing. Each W_B is open in X and hence by assumption an F_\sigma-set in X. Thus by Proposition 2, each W_B is paracompact. Note that \mathcal{W}_B is an open cover of W_B. Let \mathcal{H}_B be a locally finite open refinement of \mathcal{W}_B. Consider the following two items:

    For each j=1,2,3,\cdots, let \mathcal{V}_j=\left\{A \times B: A \in \mathcal{H}_B \text{ and } B \in \mathcal{B}_j \right\}

    \mathcal{V}=\bigcup \limits_{j=1}^\infty \mathcal{V}_j

We observe that \mathcal{V} is an open cover of X \times Y and that \mathcal{V} refines \mathcal{U}. Furthermore each \mathcal{V}_j is a locally finite collection. The open cover \mathcal{U} we start with has a \sigma-locally finite open refinement. Thus X \times Y is paracompact. \blacksquare

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Result 4

Result 4 is the statement that:

    If X is hereditarily Lindelof and Y is a separable metric space, then X \times Y is hereditarily Lindelof.

Proof of Result 4
Suppose X is hereditarily Lindelof and that Y is a separable metric space. It is well known that regular Lindelof spaces are paracompact. Thus X is paracompact. By Proposition 5, X is perfectly normal. By Result 3, X \times Y is paracompact and perfectly normal.

Let D be a countable dense subset of Y. We can think of D as a \sigma-compact space. The product of any Lindelof space with a \sigma-compact space is Lindelof (see Corollary 3 in the post “The Tube Lemma”). Thus X \times D is Lindelof. Furthermore X \times D is a dense Lindelof subspace of X \times Y. By Proposition 3, X \times Y is Lindelof. By Proposition 5, X \times Y is hereditarily Lindelof. \blacksquare

Remark
In the previous post “Bernstein Sets and the Michael Line”, a non-normal product space where one factor is Lindelof and the other factor is a separable metric space is presented. That Lindelof space is not hereditarily Lindelof (it has uncountably many isolated points). Note that by Result 4, for any such non-normal product space, the Lindelof factor cannot be hereditarily Lindelof.

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Reference

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

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\copyright \ \ 2012

Cartesian Products of Two Paracompact Spaces

In some previous posts we discuss examples surrounding the Michael line showing that the product of a paracompact space and a complete metric space needs not be normal (see “Michael Line Basics”) and that the product of a Lindelof space and a separable metric space need not be normal (see “Bernstein Sets and the Michael Line”). These examples are classic counterexamples demonstrating that both paracompactness and Lindelofness are not preserved by taking two-factor cartesian products even when one of the factors is nice (complete metric space in the first example and separable metric space in the second example). We now show some positive results. Of course, these results require additional conditions on one or both of the factors. We prove the following results.

Result 1

    If X is paracompact and Y is compact, then X \times Y is paracompact.

Result 2

    If X is paracompact and Y is \sigma-compact, then X \times Y is paracompact.

Result 3

    If X is paracompact and perfectly normal and Y is metrizable, then X \times Y is paracompact and perfectly normal.

Result 4

    If X is hereditarily Lindelof and Y is a separable metric space, then X \times Y is hereditarily Lindelof.

With Results 1 and 2, compact spaces and \sigma-compact spaces can be called productively paracompact since the product of each of these spaces with any paracompact space is paracompact. We prove Result 1 and Result 2 below.

Result 3 and Result 4 are proved in another post Cartesian Products of Two Paracompact Spaces – Continued.

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Paracompact Spaces

First, recall some definitions. All spaces are at least regular (to us regular implies Hausdorff). Let X be a space. A collection \mathcal{A} of subsets of X is said to be a cover of X if X=\bigcup \mathcal{A} (in words every point of the space belongs to one set in the collection). Furthermore, \mathcal{A} is an open cover of X is it is a cover of X consisting of open subsets of X.

Let \mathcal{A} and \mathcal{B} be covers of the space X. The cover \mathcal{B} is said to be a refinement of \mathcal{A} (\mathcal{B} is said to refine \mathcal{A}) if for every B \in \mathcal{B}, there is some A \in \mathcal{A} such that B \subset A. The cover \mathcal{B} is said to be an open refinement of \mathcal{A} if \mathcal{B} refines \mathcal{A} and \mathcal{B} is an open cover.

A collection \mathcal{A} of subsets of X is said to be a locally finite collection if for each point x \in X, there is a non-empty open subset V of X such that x \in V and V has non-empty intersection with at most finitely many sets in \mathcal{A}. An open cover \mathcal{A} of X is said to have a locally finite open refinement if there exists an open cover \mathcal{C} of X such that \mathcal{C} refines \mathcal{A} and \mathcal{C} is a locally finite collection. We have the following definition.

Definition

    The space X is said to be paracompact if every open cover of X has a locally finite open refinement.

A collection \mathcal{U} of subsets of the space X is said to be a \sigma-locally finite collection if \mathcal{U}=\bigcup \limits_{i=1}^\infty \mathcal{U}_i such that each \mathcal{U}_i is a locally finite collection of subsets of X. Consider the property that every open cover of X has a \sigma-locally finite open refinement. This on the surface is a stronger property than paracompactness. However, Theorem 1 below shows that it is actually equivalent to paracompactness. The proof of Theorem 1 can be found in [1] (Theorem 5.1.11 in page 302) or in [2] (Theorem 20.7 in page 146).

Theorem 1
Let X be a regular space. Then X is paracompact if and only if every open cover \mathcal{U} of X has a \sigma-locally finite open refinement.

Theorem 2 below is another characterization of paracompactness that is useful. For a proof of Theorem 2, see “Finite and Countable Products of the Michael Line”.

Theorem 2
Let X be a regular space. Then X is paracompact if and only if the following holds:

    For each open cover \left\{U_t: t \in T \right\} of X, there exists a locally finite open cover \left\{V_t: t \in T \right\} such that \overline{V_t} \subset U_t for each t \in T.

Theorem 3 below shows that paracompactness is hereditary with respect to F_\sigma-subsets.

Theorem 3
Every F_\sigma-subset of a paracompact space is paracompact.

Proof of Theorem 3
Let X be paracompact. Let Y \subset X such that Y=\bigcup \limits_{i=1}^\infty Y_i where each Y_i is a closed subset of X. Let \mathcal{U} be an open cover of Y. For each U \in \mathcal{U}, let U^* be open in X such that U^* \cap Y=U.

For each i, let \mathcal{U}_i^* be the set of all U^* such that U \cap Y_i \ne \varnothing. Let \mathcal{V}_i^* be a locally finite refinement of \mathcal{U}_i^* \cup \left\{X-Y_i \right\}. Let \mathcal{V}_i be the following:

    \mathcal{V}_i=\left\{V \cap Y: V \in \mathcal{V}_i^* \text{ and } V \cap Y_i \ne \varnothing \right\}

It is clear that each \mathcal{V}_i is a locally finite collection of open set in Y covering Y_i. All the \mathcal{V}_i together form a refinement of \mathcal{U}. Thus \mathcal{V}=\bigcup \limits_{i=1}^\infty \mathcal{V}_i is a \sigma-locally finite open refinement of \mathcal{U}. By Theorem 1, the F_\sigma-set Y is paracompact. \blacksquare
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Result 1

Result 1 is the statement that:

    If X is paracompact and Y is compact, then X \times Y is paracompact.

To prove Result 1, we use the Tube lemma (for a proof, see “The Tube Lemma”).

The Tube Lemma
Let X be any space and Y be compact. For each x \in X and for each open set U \subset X \times Y such that \left\{x \right\} \times Y \subset U, there is an open set O \subset X such that \left\{x \right\} \times Y \subset O \times Y \subset U.

Proof of Result 1
Let \mathcal{U} be an open cover of X \times Y. For each x \in X, choose a finite \mathcal{U}_x \subset \mathcal{U} such that \mathcal{U}_x is a cover of \left\{x \right\} \times Y. By the Tube Lemma, for each x \in X, there is an open set O_x \subset X such that \left\{x \right\} \times Y \subset O_x \times Y \subset \cup \mathcal{U}_x. Since X is paracompact, by Theorem 2, let \left\{W_x: x \in X \right\} be a locally finite open refinement of \left\{O_x: x \in X \right\} such that W_x \subset O_x for each x \in X.

Let \mathcal{W}=\left\{(W_x \times Y) \cap U: x \in X, U \in \mathcal{U}_x \right\}. We claim that \mathcal{W} is a locally finite open refinement of \mathcal{U}. First, this is an open cover of X \times Y. To see this, let (a,b) \in X \times Y. Then a \in W_x for some x \in X. Furthermore, a \in O_x and (a,b) \in \cup \mathcal{U}_x. Thus, (a,b) \in (W_x \times Y) \cap U for some U \in \mathcal{U}_x. Secondly, it is clear that \mathcal{W} is a refinement of the original cover \mathcal{U}.

It remains to show that \mathcal{W} is locally finite. To see this, let (a,b) \in X \times Y. Then there is an open V in X such that x \in V and V can meets only finitely many W_x. Then V \times Y can meet only finitely many sets in \mathcal{W}. \blacksquare

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Result 2

Result 2 is the statement that:

    If X is paracompact and Y is \sigma-compact, then X \times Y is paracompact.

Proof of Result 2
Note that the \sigma-compact space Y is Lindelof. Since regular Lindelof are normal, Y is normal and is thus completely regular. So we can embed Y into a compact space K. For example, we can let K=\beta Y, which is the Stone-Cech compactification of Y (see “Embedding Completely Regular Spaces into a Cube”). For our purpose here, any compact space containing Y will do. By Result 1, X \times K is paracompact. Note that X \times Y can be regarded as a subspace of X \times K.

Let Y=\bigcup \limits_{i=1}^\infty Y_i where each Y_i is compact in Y. Note that X \times Y=\bigcup \limits_{i=1}^\infty X \times Y_i and each X \times Y_i is a closed subset of X \times K. Thus the product X \times Y is an F_\sigma-subset of X \times K. According to Theorem 3, F_\sigma-subsets of any paracompact space is paracompact space. Thus X \times Y is paracompact. \blacksquare

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Reference

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

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\copyright \ \ 2012

Sorgenfrey Line is not a Moore Space

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

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

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

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

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Moore Spaces

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

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

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

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

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

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Sorgenfrey Line

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

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

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

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

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Remark

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

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Reference

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

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\copyright \ \ 2012

A Space with G-delta Diagonal that is not Submetrizable

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

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

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

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

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

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

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

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

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

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

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Moore Space

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

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

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

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

Reference

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

When is a Pseudocompact Space Metrizable?

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

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

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

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

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

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

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

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

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

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

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Pseudocompact Spaces

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Reference

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

Elementary Examples of Lindelof Spaces and Separable Spaces

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

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

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

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

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

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

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

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

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

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Remark

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

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

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

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

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

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

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

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

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