Category Archives: Advanced Topology

A note on metrization theorems for compact spaces

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

On Spaces That Can Never Be Dowker

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A Dowker space is a normal space X for which the product with the closed unit interval [0,1] is not normal. In 1951, Dowker characterized Dowker’s spaces as those spaces that are normal but not countably paracompact ([1]). Soon after, spaces that are normal but not countably paracompact became known as Dowker spaces. In 1971, M. E. Rudin ([2]) constructed a ZFC example of a Dowker’s space. But this Dowker’s space is large. It has cardinality (\omega_\omega)^\omega and is pathological in many ways. Thus the search for “nice” Dowker’s spaces continued. The Dowker’s spaces being sought were those with additional properties such as having various cardinal functions (e.g. density, character and weight) countable. Many “nice” Dowker’s spaces had been constructed using various additional set-theoretic assumptions. In 1996, Balogh constructed a first “small” Dowker’s space (cardinaltiy continuum) without additional set-theoretic axioms beyond ZFC ([4]). Rudin’s survey article is an excellent reference for Dowker’s spaces ([3]).

In this note, I make several additional observations on Dowker’s spaces. In this previous post, I presented a proof of the Dowker’s theorem characterizing the normal spaces for which the product with the unit interval is normal (see the statement of the Dowker’s theorem below). In another post, I showed that perfectly normal spaces can never be Dowker’s spaces. Based on the Dowker’s theorem, several other classes of spaces are easily seen as not Dowker.

Dowker’s Theorem. For a normal space X, the following conditions are equivalent.

  1. The space X is countably paracompact.
  2. The product X \times Y is normal for any infinite compact metric space Y.
  3. The product X \times [0,1] is normal.
  4. For each sequence of closed subsets \lbrace{A_0,A_1,A_2,...}\rbrace of X such that A_0 \supset A_1 \supset A_2 \supset ... and \bigcap_{n<\omega} A_n=\phi, there is open sets U_n \supset A_n for each n such that \bigcap_{n<\omega} U_n=\phi.

Observations. If X is perfectly compact, then it can be shown that it is countably paracompact by showing that it satisfies condition 4 in the Dowker’s theorem (there is a proof in this blog). Thus there are no perfectly normal Dowker’s spaces. There are no countably compact Dowker’s spaces since any countably compact space is countably paracompact. This can also be seen using condition 4 above. In a countably compact space, any decreasing nested sequence of closed sets has non-empty intersection and thus condition 4 is satisfied vacuously. Furthermore, all metric spaces, compact spaces, regular Lindelof spaces cannot be Dowker since these spaces are paracomapct.

Normal Moore spaces are perfectly normal. Thus there are no Dowker’s spaces that are Moore spaces. Note that a space is perfectly normal if it is normal and if every closed set is G_\delta. We show that in a Moore space, every closed set is G_\delta. Let \lbrace{\mathcal{O}_n:n \in \omega}\rbrace be a development for the regular space X. Let A be a closed set in X. We show that A is a G_\delta- set in X. For each n, let U_n=\lbrace{O \in \mathcal{O}_n:O \bigcap A \neq \phi}\rbrace. Obviously, A \subset \bigcap_n U_n. Let x \in \bigcap_n U_n. If x \notin A, there is some n such that for each O \in \mathcal{O}_n with x \in O, we have O \subset X-A. Since x \in \bigcap_n U_n, x \in O for some O \in \mathcal{O}_n and O \cap A \neq \phi, a contradiction. Thus we have A=\bigcap_n U_n.

There are other classes of spaces that can never be Dowker. We point these out without proof. For example, there are no linearly ordered Dowker’s spaces and there are no monotonically normal Dowker’s spaces (see Rudin’s survey article [3]).

Reference

  1. Dowker, C. H., On Countably Paracompact Spaces, Canad. J. Math. 3, (1951) 219-224.
  2. Rudin, M. E., A normal space X for which X \times I is not normal, Fund. Math., 73 (1971), 179-186.
  3. Rudin, M. E., Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 761-780.
  4. Balogh, Z., A small Dowker space in ZFC, Proc. Amer. Math. Soc., 124 (1996), 2555-2560.

Countably Compact Spaces with G-delta Diagonals

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It is a classic result in general topology that any compact space with a G_\delta-diagonal is metrizable ([3]). This theorem also holds for countably compact spaces (due to Chaber in [2]). The goal of this post is to present a proof of this theorem. We prove that if X is countably compact and has a G_\delta-diagonal, then X is compact and thus metrizable. All spaces are at least Hausdorff. This post has a discussion on the theorem on compact spaces with G_\delta-diagonal. This post has a discussion on some metrizaton theorems for compact spaces.

If \mathcal{T} is a collection of subsets of a space X, then for each x \in X, define st(x,\mathcal{T})=\bigcup\lbrace{T \in \mathcal{T}:x \in T}\rbrace. A sequence of open covers \lbrace{\mathcal{T}_n:n \in \omega}\rbrace of the space X is a G_\delta-diagonal sequence for X if for each x \in X, we have \lbrace{x}\rbrace=\bigcap_{n<\omega} st(x,\mathcal{T}_n). We use the following lemma (due to Ceder, [1]). This lemma was proved in this previous post.

Lemma. The space X has a G_\delta-diagonal if and only if it has a G_\delta-diagonal sequence.

Theorem. Let X be a countably compact space that has a G_\delta-diagonal. Then X is compact.

Proof. Let X be a countably compact space. Let \lbrace{\mathcal{T}_n:n \in \omega}\rbrace be a G_\delta-diagonal sequence for X. If X is Lindelof, then we are done. Suppose we have an open cover \mathcal{V} of X that has no countable subcover. From this open cover \mathcal{V}, we derive a contradiction.

We inductively, for each \alpha < \omega_1, choose a point x_\alpha \in X and an integer m(\alpha) \in \omega with the following properties:

For each \alpha < \omega_1,

  1. x_\alpha \in X-\bigcup\lbrace{st(x_\beta,\mathcal{T}_{m(\beta)}): \beta < \alpha}\rbrace, and
  2. the open cover \mathcal{V} does not have a countable subcollection that covers X-\bigcup_{\beta \leq \alpha} st(x_\beta,\mathcal{T}_{m(\beta)}).

To start off, choose x_0 \in X. There is an integer m(0) \in \omega such that no countable subcollection of \mathcal{V} covers X-st(x_0,\mathcal{T}_{m(0)}). Suppose this integer m(0) does not exist. Then for each n \in \omega, we have a countable \mathcal{V}_n \subset \mathcal{V} such that \mathcal{V}_n covers X-st(x_0,\mathcal{T}_n). Then \bigcup_{n<\omega} \mathcal{V}_n would be a countable subcollection of \mathcal{V} that covers X-\lbrace{x_0}\rbrace. This would mean that \mathcal{V} has a countable subcover of X.

Suppose that \lbrace{x_\beta:\beta<\alpha}\rbrace and \lbrace{m(\beta):\beta<\alpha}\rbrace have been chosen such that conditions (1) and (2) are satisfied for each \beta<\alpha. We have the following claim. Proving this claim allows us to choose x_\alpha and m(\alpha).

Claim. No countable subcollection of \mathcal{V} covers X-\bigcup_{\beta<\alpha} st(x_\beta,\mathcal{T}_{m(\beta)}).

Suppose we do have a countable \mathcal{W} \subset \mathcal{V} such that \mathcal{W} covers X-\bigcup_{\beta<\alpha} st(x_\beta,\mathcal{T}_{m(\beta)}). Then \mathcal{S}=\lbrace{st(x_\beta,\mathcal{T}_m(\beta)):\beta < \alpha}\rbrace \cup \mathcal{W} is a countable open cover of X and thus has a finite subcover \mathcal{F}. Let \delta be the largest ordinal <\alpha such that st(x_\delta,\mathcal{T}_m(\delta)) is in this finite subcover \mathcal{F}. Then \mathcal{W} is a counntable subcollection of \mathcal{V} that covers X-\bigcup_{\beta \leq \delta} st(x_\beta,\mathcal{T}_{m(\beta)}). This violates condition (2) above for the ordinal \delta. This proves the claim.

Now, pick x_\alpha \in X-\bigcup\lbrace{st(x_\beta,\mathcal{T}_{m(\beta)}): \beta < \alpha}\rbrace. There must be some integer m(\alpha) \in \omega such that conditon (2) above is satisfied for \alpha. If not, for each n \in \omega, there is some countable \mathcal{V}_n \subset \mathcal{V} such that \mathcal{V}_n covers X-\bigcup_{\beta \leq \alpha} st(x_\beta,\mathcal{T}_n). Then \bigcup_{n<\omega} \mathcal{V}_n would be a countable subcollection of \mathcal{V} that covers X-\biggl(\bigcup_{\beta<\alpha} st(x_\beta,\mathcal{T}_{m(\beta)}) \biggr) \bigcup \lbrace{x_\alpha}\rbrace. This would mean that \mathcal{V} has a countable subcover of X-\bigcup_{\beta<\alpha} st(x_\beta,\mathcal{T}_{m(\beta)}). This violates the above claim. Now the induction process is completed.

To conclude the proof of the theorem, note that there is some n \in \omega and there is some uncountable D \subset \omega_1 such that for each \alpha \in D, n=m(\alpha). Then Y=\lbrace{x_\alpha:\alpha \in D}\rbrace is an uncountable closed and discrete set in X. Note that each open set in \mathcal{T}_n contains at most one point of Y. Thus X must be Lindelof. With X being countably compact, X is compact.

Reference

  1. Ceder, J. G. Some generalizations of metric spaces, Pacific J. Math., 11 (1961), 105-125.
  2. Chaber, Conditions which imply compactness in countably compact spaces, Bull. Acad. Pol. Sci. Ser. Math., 24 (1976), 993-998.
  3. Sneider, V., Continuous images of Souslin and Borel sets: metrization theorems, Dokl. Acad. Nauk USSR, 50 (1945), 77-79.

Metrization Theorems for Compact Spaces

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In this blog I have already presented two metrization theorems for compact spaces: (1) any compact space with a countable network is metrizable (see the post), (2) any compact space with a G_\delta-diagonal is metrizable (see the post). I now present another classic theorem: any countably compact space with a point-countable base is metrizable. This theorem is a classic result of Miscenko ([1]). All spaces are at least Hausdorff and regular. We have the following three metrization theorems for compact spaces. In subsequent posts, I will discuss generalizations of these theorems and discuss related concepts.

Thoerem 1. Any compact space with a countable network is metrizable.
The proof is in this post.

Thoerem 2. Any compact space with a G_\delta-\text{diagonal} is metrizable.
The proof is in this post.

Thoerem 3. Any countably compact space with a point-countable base is metrizable.

A base \mathcal{B} for a space X is a point-countabe base if every point in X belongs to at most countably elements of \mathcal{B}.

Proof of Theorem 3. Let \mathcal{B} be a point-countable base for the countably compact space X. We show that X is separable. Once we have a countable dense subset, the base \mathcal{B} has to be a countable base. So we inductively define a sequence of countable sets \lbrace{D_0,D_1,...}\rbrace such that D=\bigcup_{n<\omega}D_n is dense in X.

Let D_0=\lbrace{x_0}\rbrace be a one-point set to start with. For n>0, let E_n=\bigcup_{i<n}D_n. Let \mathcal{B}_n=\lbrace{B \in \mathcal{B}:B \cap E_n \neq \phi}\rbrace. For each finite T \subset \mathcal{B}_n such that X - \bigcap T \neq \phi, choose a point x(T) \in X - \bigcup T. Let D_n be the union of E_n and the set of all points x(T). Let D=\bigcup_{n<\omega}D_n.

We claim that \overline{D}=X. Suppose we have x \in X-\overline{D}. Let \mathcal{A}=\lbrace{B \in \mathcal{B}:B \cap D \neq \phi \phantom{X} \text{and} \thinspace x \notin B}\rbrace. We know that \mathcal{A} is countable since every element of \mathcal{A} contains points of the countable set D. We also know that \mathcal{A} is an open cover of \overline{D}. By the countably compactness of \overline{D}, we can find a finite T \subset \mathcal{A} such that \overline{D} \subset \bigcup T. The finite set T must have appeared during the induction process of selecting points for D_n for some n (i.e. T \subset \mathcal{B}_n). So a point x(T) has been chosen such that x(T) \notin \bigcup T (thus we have x(T) \in D_n \subset \overline{D}). On the other hand, since \overline{D} \subset \bigcup T, we observe that x(T) \notin \overline{D}, producing a contradiction. Thus the countable set D is dense in X, making the point-countable base \mathcal{B} a countable base.

Reference

  1. Miscenko, A., Spaces with a point-countable base, Dokl. Acad. Nauk SSSR, 144 (1962), 985-988. (English translation: Soviet Math. Dokl. 3 (1962), 1199-1202).

Compact Spaces With G-delta Diagonals

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In a previous post, I showed that any compact space with a countable network is metrizable. Another classic metrization theorem for compact spaces is that any compact space with a G_\delta-diagonal is metrizable ([6]). The theroem I try to prove is: for a compact space X, X^2 is perfectly normal if and only if X has a G_\delta-diagonal if and only if X is metrizable. My proof is based on the notion of G^*_\delta-diagonal. Every compact space with a G_\delta-diagonal has a G^*_\delta-diagonal, which allows us to define a countable base. The theorem discussed here had been generalized (see the comment at the end of this post). All spaces are at least Hausdorff.

Let X be a space. The set \Delta=\lbrace{(x,x):x \in X}\rbrace is called the diagonal of the space X. The space X has a G_\delta-diagonal if \Delta is a G_\delta-set in X^2.

Let \mathcal{G} be a collection of subsets of X and let x \in X. Define st(x,\mathcal{G})=\bigcup \lbrace{G \in \mathcal{G}:x \in G}\rbrace. A sequence \lbrace{\mathcal{G}_n}\rbrace_{n<\omega} of open covers of X is called a G_\delta-diagonal sequence of X if for each x \in X, \lbrace{x}\rbrace=\bigcap_{n<\omega} st(x,\mathcal{G}_n). Lemma 1 shows that a space has a G_\delta-diagonal if and only if it has a G_\delta-diagonal sequence. This lemma is due to Ceder ([2]).

Another notion we need is that of the G^*_\delta-diagonal. The space X has a G^*_\delta-diagonal if there is a G_\delta-diagonal sequence \lbrace{\mathcal{G}_n}\rbrace_{n<\omega} such that for each x \in X, \lbrace{x}\rbrace=\bigcap_{n<\omega} \overline{st(x,\mathcal{G}_n)}. Such a G_\delta-diagonal sequence is called a G^*_\delta-diagonal sequence. The notion of G^*_\delta-diagonal is due to R. E. Hodel ([4]). Lemma 2 below shows that any compact space with a G_\delta-diagonal has a G^*_\delta-diagonal.

Lemma 1. The space X has a G_\delta-diagonal if and only if it has a G_\delta-diagonal sequence.

Proof. \Rightarrow Suppose that \Delta=\bigcap_{n<\omega}U_n where each U_n is open in X^2. Let \tau denote the topology on X. For each n, let \mathcal{G}_n=\lbrace{V \in \tau:V \times V \subset U_n}\rbrace. We claim that for each x \in X, \lbrace{x}\rbrace=\bigcap_{n<\omega} st(x,\mathcal{G}_n). Obviously \lbrace{x}\rbrace \subset \bigcap_{n<\omega} st(x,\mathcal{G}_n). Let y \in \bigcap_{n<\omega} st(x,\mathcal{G}_n). For each n, y \in V_n where V_n \in \mathcal{G}_n and x \in V_n. Thus (x,y) \in V_n \times V_n \subset U_n for each n. This implies (x,y) \in \Delta and x=y. Thus \lbrace{x}\rbrace=\bigcap_{n<\omega} st(x,\mathcal{G}_n). We have established that \{ \mathcal{G}_n \} is a G_\delta-diagonal sequence of X.

\Leftarrow Suppose \lbrace{\mathcal{G}_n}\rbrace is a G_\delta-diagonal sequence of X. For each n, let U_n=\bigcup \lbrace{V \times V:V \in \mathcal{G}_n}\rbrace. Since \bigcap_n st(x,\mathcal{G}_n)=\{ x \}, \Delta \subset \bigcap_{n<\omega} U_n. To show the set inclusion for the other direction, 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{G}_n. This implies that y \in st(x,\mathcal{G}_n) for each n. It follows that y=x. Thus \Delta = \bigcap_{n<\omega} U_n.

Lemma 2. If X is compact and has a G_\delta-diagonal, then X has a G^*_\delta-diagonal. Furthermore, each open cover in the G_\delta-diagonal sequence is finite.

Proof. Let \lbrace{\mathcal{G}_n}\rbrace_{n<\omega} be the G_\delta-diagonal sequence obtained in Lemma 1. We inductively define \lbrace{\mathcal{H}_n}\rbrace_{n<\omega}, another G_\delta-diagonal sequence.

Using the compactness of X, obtain a finite subcollection \mathcal{H}_0 of \mathcal{G}_0 such that \mathcal{H}_0 is a cover of X. Here’s how I obtain \mathcal{H}_1. For each x \in X, choose an open set G_x \in \mathcal{G}_1 and an open set H_x \in \mathcal{H}_0 such that x \in G_x and x \in H_x. Choose open set V_x such that x \in V_x and \overline{V_x} \subset G_x \cap H_x. Let \mathcal{H}_1 be a finite subcollection of \lbrace{V_x:x \in X}\rbrace such that \mathcal{H}_1 is a cover of X. Continue the inductive process and we produce a sequence of open covers \lbrace{\mathcal{H}_n}\rbrace_{n<\omega} satisfying the following two claims.

Claim 1
For each x \in X, \lbrace{x}\rbrace=\bigcap_{n<\omega} st(x,\mathcal{H}_n).

Because each open cover \mathcal{H}_n is chosen to be a subcover or a refinement of the open cover \mathcal{G}_n, we have st(x,\mathcal{H}_n) \subset st(x,\mathcal{G}_n). Since \lbrace{x}\rbrace=\bigcap_{n<\omega} st(x,\mathcal{G}_n) (from the definition of G_\delta-diagonal sequence), we have \lbrace{x}\rbrace=\bigcap_{n<\omega} st(x,\mathcal{H}_n).

Claim 2
For each x \in X, \lbrace{x}\rbrace=\bigcap_{n<\omega} \overline{st(x,\mathcal{H}_n)}.

We only need to show \bigcap_{n<\omega} \overline{st(x,\mathcal{H}_n)} \subset \lbrace{x}\rbrace. Let y \in \overline{st(x,\mathcal{H}_n)} for each n. Because \mathcal{H}_n is finite, y \in \overline{V} for some V \in \mathcal{H}_n with x \in V. Each such \overline{V} \subset U for some U \in \mathcal{H}_{n-1}. Thus y \in \overline{st(x,\mathcal{H}_n)} implies y \in st(x,\mathcal{H}_{n-1}). By Claim 1, y=x.

We have shown that X has a G^*_\delta-diagonal by producing a G^*_\delta-diagonal sequence \lbrace{\mathcal{H}_n}\rbrace_{n<\omega}.

Theorem. If X is compact and has a G_\delta-diagonal, then X is metrizable.

Proof. Let \lbrace{\mathcal{H}_n}\rbrace_{n<\omega} be the G^*_\delta-diagonal sequence obtained in Lemma 2. Furthermore, each \mathcal{H}_n is a finite open cover. Let \mathcal{H}=\bigcup_{n<\omega} \mathcal{H}_n. The collection \mathcal{H} satisfies the properties stated in the following two claims.

Claim 3
For each x,y \in X with x \neq y, there is a U \in \mathcal{H} such that x \in U and y \notin \overline{U}.

Since \lbrace{x}\rbrace=\bigcap_{n<\omega} \overline{st(x,\mathcal{H}_n)}, y \notin \overline{st(x,\mathcal{H}_n)} for some n. Then there is some U \in \mathcal{H}_n such that x \in U and y \notin \overline{U}. In fact, for any U \in \mathcal{H}_n with x \in U, it must be the case that y \notin \overline{U}.

Claim 4
Let \mathcal{B}=\lbrace{X-\overline{\bigcup F}: F \subset \mathcal{H} \phantom{x} and \phantom{x} \vert F \lvert < \omega}\rbrace. Then \mathcal{B} is a countable base for X.

To see Claim 4, note that \mathcal{B} is a cover of X and is closed under finite intersections. This makes \mathcal{B} is base for a topology. To show that this base generate the same topology on X, let y \in U \subset X where U is open. Then X-U is compact. For each x \in X-U, let V_x \in \mathcal{H} such that x \in V_x and y \notin \overline{V_x}. We can choose F=\lbrace{V_{x(0)},...,V_{x(n)}}\rbrace such that F is a cover of X-U. Then y \in X-\overline{\bigcup F} \subset U.

With Claims 3 and 4, the theorem is established.

Corollary. Let X be a compact space. The following conditions are equivalent.

  1. X^2 is perfectly normal.
  2. X has a G_\delta-diagonal.
  3. X has a countable base.

Proof. 1 \rightarrow 2 and 3 \rightarrow 1 are obvious. 2 \rightarrow 3 follows from the theorem.

Examples. Based on the corollary, any non-metrizable compact Hausdorff space does not have a G_\delta-diagonal. One handy example is the uncountable product of the unit interval I^{\omega_1} where I=[0,1]. Both I \times I with the lexicographic order and the double arrow space are compact and non-metrizable (thus have no G_\delta-diagonal). I discussed these two spaces in a previous post.

Comment. The notion of G_\delta-diagonal plays an important role in metrization theorems. The theorem for compact space with G_\delta-diagonal had long been generalized. For example, in [3] Chaber had shown that any countably compact space with a G_\delta- diagonal is metrizable. In [1] and [5], it was shown that any paracompact space with a G_\delta-diagonal is submetrizable. The theorem proved in this post would simply be a corollary of this result. In upcoming posts, I plan to discuss some of these theorems as well as explore the connection of submetrizability and various G_\delta-diagonal properties.

Reference

  1. Borges, C. R. On stratifiable spaces, Pacific J. Math., 17 (1966), 1-16.
  2. Ceder, J. G. Some generalizations of metric spaces, Pacific J. Math., 11 (1961), 105-125.
  3. Chaber, Conditions which imply compactness in countably compact spaces, Bull. Acad. Pol. Sci. Ser. Math., 24 (1976), 993-998.
  4. Hodel, R., E., Moore spaces and w \Delta-spaces, Pacific J. Math., 38, (1971), 641-652.
  5. Okuyama, A., On metrizability of M-spaces, Proc. Japan. Acad., 40, 176-179.
  6. Sneider, V., Continuous images of Souslin and Borel sets: metrization theorems, Dokl. Acad. Nauk USSR, 50 (1945), 77-79.

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Perfectly Normal Spaces Can Never Be Dowker Spaces

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The Dowker’s theorem states that for a normal space X, X \times [0,1] is normal if and only if X is countably paracompact. Since this theorem was published, any normal space that is not countably paracompact became known as Dowker space. There are classes of spaces that can never be Dowker spaces (e.g. metrizable spaces, paracompact spaces, compact spaces and Lindelof spaces). In [Katetov], it was shown that there are no perfectly normal Dowker spaces. My blog has a proof of the Dowker’s theorem (see the proof here). For more background on Dowker’s spaces, see the survey article [Rudin]. Dowker’s theorem was published in [Dowker].

Theorem. If X is perfectly normal, then X is countably paracompact.

To prove this theorem, we use the following characterization of countably paracompactness (you can find a proof here).

Lemma. Let X be a normal space. Then X is countably paracompact if and only if for each sequence \lbrace{A_n:n \in \omega}\rbrace of closed subsets of X such that A_0 \supset A_1 \supset ... and \bigcap_n A_n=\phi, there exist open sets B_n \supset A_n such that \bigcap_n B_n=\phi.

Proof of Theorem. Suppose X is perfectly normal. Let A_0 \supset A_1 \supset ... be a sequence of closed sets such that \bigcap_n A_n=\phi. For each n, let A_n=\bigcap_{i<\omega} U_{n,i} where each U_{n,i} is open in X. For each n, define B_n=\bigcap_{i, j \leq n}U_{i,j}. Clearly, B_n \supset A_n. It is easy to see that \bigcap_n B_n=\phi. Note that all the open sets U_{n,j} are used in defining the sequence B_0,B_1,B_2,\cdots. Thus \bigcap_n B_n \neq \phi would imply \bigcap_n A_n \neq \phi.

Comment. As a consequence of this theorem and the Dowker’s theorem, if X is perfectly normal, then X \times Y is normal for any compact metric space Y.

Reference
[Dowker]
Dowker, C. H. [1951], On Countably Paracompact Spaces, Canad. J. Math. 3, 219-224.

[Katetov]
Katetov, M., On real-valued functions in topological spaces, Fund. Math. 38 (1951), pp. 85-91.

[Rudin]
Rudin, M. E., [1984], Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 761-780.

____________________________________________________________________
\copyright \ 2009-2016 \text{ by Dan Ma}
Revised November 23, 2016

A Theorem About Hereditarily Normality

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Let Y be either the closed unit interval \mathbb{I}=[0,1] or [0,\omega]=\omega+1. Let X be any one of the following spaces:

X=[0,\omega_1]=\omega_1+1,
X=[0,\omega_1)=\omega_1,
X= Michael Line,
X= Sorgenfrey Line.

The cross product X \times Y is normal in all four cases. The X factor, in the order listed, is compact, countably compact, paracompact and Lindelof. Thus they are all countably paracompact. According to the Dowker’s theorem, the product of a normal space X and a compact metric space Y is normal if and only if X is countably paracompact. My goal here is to show that the four cases of X \times Y here cannot be hereditarily normal. This is from a theorem due to Katetov. We prove the following theorem.

Theorem. If X \times Y is hereditarily normal, then either X is perfectly normal or every countably infinite subspace of Y is closed and discrete.

Proof. Suppose that X is not perfectly normal and Y has a countably infinite subset that is not closed and discrete. Let H \subset X be a closed set that is not a G_\delta-set. Let C=\lbrace{y_n:n \in \omega}\rbrace \subset Y be an infinite set with an accumulation point y. We assume that y \notin C.

We show that the open subspace U=(X \times Y)-(H \times \lbrace{y}\rbrace) is not normal. To this end, let A=H \times (Y-\lbrace{y}\rbrace) and B=(X-H) \times \lbrace{y}\rbrace. The sets A and B are disjoint closed subspaces of the open subspace U. Suppose we have disjoint open sets S and T such that A \subset S and B \subset T.

For each n \in \omega, let O_n=\lbrace{x \in X:(x,y_n) \in S}\rbrace. Each O_n is open and H \subset O_n. Thus H \subset \bigcap_n O_n. Let x \in \bigcap_n O_n. Then (x,y) \in \overline{S}. This means x \in H. If x \notin H, then (x,y) \in B \subset T (which is impossible). So we have H=\bigcap_n O_n, indicating that H is a G_\delta-set, and leading to a contradiction. So the subspace U=(X \times Y)-(H \times \lbrace{y}\rbrace) is not normal.

Corollary. For countably compact spaces (in particular compact spaces) X and Y, if the Cartesian product X \times Y is hereditarily normal, then both X and Y are prefectly normal.

Proof. Note that both factors, being countably compact, cannot have closed and discrete infinite subsets.

Comment. Note that the converse of the corollary is not true. Let both factors be the double arrow space, which is perfectly normal. But the square of the double arrow space contains a copy of the Sorgenfrey Plane, which is not normal.

Reference
[Katetov] Katetov, M., [1948] Complete normality of Cartesian products, Fund. Math., 36, 271-274.

 

Dowker’s Theorem

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Let \mathbb{I} be the closed unit interval [0,1]. In 1951, Dowker proved that for normal spaces X, X \times \mathbb{I} is normal if and only if X is countably paracompact. A space X is countably paracompact if every countable open cover of X has a locally finite open refinement. Dowker’s theorem is a fundamental result on products of normal spaces. With this theorem, a question was raised about the existence of a normal but not countably paracompact space (such a space became known as a Dowker space). For a detailed discussion on Dowker spaces, see the survey paper [Rudin]. The focus here is on presenting a proof for the Dowker’s theorem, laying the groundwork for future discussion.

An open cover \mathcal{V}=\lbrace{V_\alpha:\alpha \in S}\rbrace of a space X is shrinkable if there exists an open cover\mathcal{W}=\lbrace{W_\alpha:\alpha \in S}\rbrace such that \overline{W_\alpha} \subset V_\alpha for each \alpha \in S. The open cover \mathcal{W} is called a shrinking of \mathcal{V}. We use the following lemma in proving Dowker’s theorem. Go here to see a proof of this lemma.

Lemma
A space X is normal if and only if every point-finite open cover of X is shrinkable.

Dowker’s Theorem
For a normal space X, the following conditions are equivalent:

  1. X is a countably paracompact space.
  2. If \lbrace{U_n:n<\omega}\rbrace is an open cover of X, then there is a locally finite open refinement \lbrace{V_n:n<\omega}\rbrace such that \overline{V_n} \subset U_n for each n.
  3. X \times Y is normal for any compact metrizable space Y.
  4. X \times \mathbb{I} is normal.
  5. For each sequence of closed sets such that A_0 \supset A_1 \supset ... and \cap_n A_n=\phi, there exists open sets B_n \supset A_n such that \cap_n B_n=\phi.

Proof
1 \Longrightarrow 2
Suppose X is countably paracompact. Let \mathcal{U}=\lbrace{U_n:n<\omega}\rbrace be an open cover of X. Then \mathcal{U} has a locally open refinement \mathcal{V}. For each n<\omega, let W_n=\bigcup \lbrace{V \in \mathcal{V}:V \subset U_n}\rbrace. Note that \lbrace{W_n:n<\omega}\rbrace is still locally finite (thus point-finite). By the lemma, it has a shrinking \lbrace{V_n:n<\omega}\rbrace.

2 \Longrightarrow 3
Let Y be a compact metrizable space. Let \lbrace{E_0,E_1,E_2,...}\rbrace be a base of Y such that it is closed under finite unions. Let H,K be two disjoint closed subsets of X \times Y. The goal here is to find an open set \mathcal{V} such that H \subset \mathcal{V} and \overline{\mathcal{V}} misses K.

For each x \in X, consider the following:
S_x=\lbrace{y \in Y: (x,y) \in H}\rbrace
T_x=\lbrace{y \in Y: (x,y) \in K}\rbrace.

For each n \in \omega, let O_n be defined by:
O_n=\lbrace{x \in X:S_x \subset E_n} and T_x \subset Y-\overline{E_n}\rbrace.

Claim 1.
Each x \in X is an element of some O_n.

Both S_x and T_x are compact. Thus we can find some E_n such that S_x \subset E_n and T_x \subset Y-\overline{E_n}.

Claim 2.
Each O_n is open in X.

Fix z \in O_n. For each y \in Y-E_n, (z,y) \notin H. Choose open set A_y \times B_y such that (z,y) \in A_y \times B_y and A_y \times B_y misses H. The set of all B_y is a cover of Y-E_n. We can find a finite subcover covering Y-E_n, say B_{y(0)},B_{y(1)},...,B_{y(k)}. Let A=\bigcap_i A_{y(i)}.

For each a \in \overline{E_n}, (z,a) \notin K. Choose open set C_a \times D_a such that (z,a) \in C_a \times D_a and C_a \times D_a misses K. The set of all D_a is a cover of \overline{E_n}. We can find a finite subcover covering \overline{E_n}, say D_{a(0)},D_{a(1)},...,D_{a(j)}. Let C=\bigcap_i C_{a(i)}.

Now, let O=A \cap C. Clearly z \in O. To show that O \subset O_n, pick x \in O. We want to show S_x \subset E_n and T_x \subset Y-\overline{E_n}. Suppose we have y \in S_x and y \in Y-E_n. Then y \in B_{y(i)} for some i. As a result, (x,y) \in A_{y(i)} \times B_{y(i)}. This would mean that (x,y) \notin H. But this contradicts with y \in S_x. So we have S_x \subset E_n. On the other hand, suppose we have y \in T_x and y \in \overline{E_n}. Then y \in D_{a(i)} for some i. As a result, (x,y) \in C_{a(i)} \times D_{a(i)}. This means that (x,y) \notin K. This contradicts with y \in T_x. So we have T_x \subset Y-\overline{E_n}. It follows that O \subset O_n and O_n is open in X.

Claim 3.
We can find an open set \mathcal{V} such that H \subset \mathcal{V} and \overline{\mathcal{V}} misses K.

Let \mathcal{O}=\bigcup_n O_n \times \overline{E_n}. Note that H \subset \mathcal{O} and \mathcal{O} misses K. The open set \mathcal{V} being constructed will satisfiy H \subset \mathcal{V} \subset \overline{\mathcal{V}} \subset \mathcal{O}. The open cover \lbrace{O_n}\rbrace has a locally finite open refinement \lbrace{V_n}\rbrace such that \overline{V_n} \subset O_n for each n. Now define \mathcal{V}=\bigcup_n V_n \times E_n. Note that H \subset \mathcal{V}.

We also have \overline{\mathcal{V}}=\overline{\bigcup_n V_n \times E_n}=\bigcup_n \overline{V_n \times E_n}. We have a closure preserving situation because the open sets V_n are from a locally finite collection. Continue the derivation and we have:

=\bigcup_n \overline{V_n} \times \overline{E_n} \subset \bigcup_n O_n \times \overline{E_n}=\mathcal{O}.

3 \Longrightarrow 4 is obvious.

4 \Longrightarrow 5
Let A_0 \supset A_1 \supset ... be closed sets with empty intersection. We show that A_n can be expanded by open sets B_n such that A_n \subset B_n and \bigcap_n B_n=\phi.

Choose p \in \mathbb{I} and a sequence of distinct points p_n \in \mathbb{I} converging to p. Let H=\cup \lbrace{A_n \times \lbrace{p_n}\rbrace:n \in \omega}\rbrace and K=X \times \lbrace{p}\rbrace. These are disjoint closed sets. Since X \times \mathbb{I} is normal, we can find open set V \subset X \times \mathbb{I} such that H \subset V and \overline{V} misses K.

Let B_n=\lbrace{x \in X:(x,p_n) \in V}\rbrace. Note that each B_n is open in X. Also, A_n \subset B_n. We want to show that \bigcap_n B_n=\phi. Let x \in X. The point (x,p) \in K and (x,p) \notin \overline{V}. There exist open E \subset X and open F \subset Y such that (x,p) \in E \times F and (E \times F) \cap \overline{V}=\phi. Then p_n \in F for some n. Note that (x,p_n) \notin V. Thus x \notin B_n. It follows that \bigcap_n B_n=\phi.

5 \Longrightarrow 1
Let \lbrace{T_n:n \in \omega}\rbrace be an open cover of X. Let E_n=\bigcup_{i \leq n} T_i. Let A_n=X-E_n. Each A_n is closed and \bigcap_n A_n=\phi. So there exist open sets B_n \supset A_n such that \bigcap_n B_n=\phi.

Since X is normal, there exists open W_n such that A_n \subset W_n \subset \overline{W_n} \subset B_n. It follows that \bigcap_n \overline{W_n}=\phi. Let O_n=X-\bigcap_{i \leq n} \overline{W_n}.

Claim. \lbrace{O_n:n \in \omega}\rbrace is an open cover of X. Furthermore, \overline{O_n} \subset \bigcup_{i \leq n}T_i.
Let x \in X. Since \bigcap_n B_n=\phi, x \notin B_n for some n. Then x \notin \overline{W_n} and x \in O_n. To show the second half of the claim, let y \notin \bigcup_{i \leq n}T_i. Then y \notin E_i for each i \leq n. This means y \in A_i and y \in W_i for each i \leq n. Then \bigcap_{i \leq n}W_i is an open set containing y that contains no point of O_n. Thus we have \overline{O_n} \subset \bigcup_{i \leq n}T_i.

Let S_n=T_n-\bigcup_{i < n} \overline{O_i}. We show that \lbrace{S_n:n \in \omega}\rbrace is a locally finite open refinement of \lbrace{T_n:n \in \omega}\rbrace. Clearly S_n \subset T_n. To show that the open sets S_n form a cover, let x \in X. Choose least n such that x \in T_n. Then for each i<n, x \notin E_i. This means x \in A_i and x \in \overline{W_i} for each i<n. It follows that x \notin O_i for each i<n. This implies x \in S_n. To see that the open sets S_n form a locally finite collection, note that each x \in X belong to some O_n. The open set O_n misses S_m for all m>n. Thus O_n can only meet S_i for i \leq n. We just prove that X is countably paracompact.

Note. In 4 \Longrightarrow 5, all we need is that the factor Y has a non-trivial convergent sequence. That is, if X \times Y is normal and Y has a non-trivial convergent sequence, then X satisfies condition 5.

Reference
[Dowker] Dowker, C. H. [1951], On Countably Paracompact Spaces, Canad. J. Math. 3, 219-224.

[Rudin] Rudin, M. E., [1984], Dowker Spaces, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 761-780.

____________________________________________________________________
\copyright \ 2009-2016 \text{ by Dan Ma}
Revised November 27, 2016