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

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2 thoughts on “Cartesian Products of Two Paracompact Spaces

  1. Pingback: The product of locally compact paracompact spaces | Dan Ma's Topology Blog

  2. Pingback: Extracting more information from Dowker’s theorem | Dan Ma's Topology Blog

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