One way to find collectionwise normal spaces

Collectionwise normality is a property that is weaker than paracompactness and stronger than normality (see the implications below). Normal spaces need not be collectionwise normal. Bing’s Example G is an example of a normal and not collectionwise normal space (see the blog post “Bing’s Example G”). We discuss one instance when normal spaces are collectionwise normal, giving a way to obtain collectionwise normal spaces that are not paracompact.

    \text{ }
    \text{paracompact} \Longrightarrow \text{collectionwise normal} \Longrightarrow \text{normal}

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Collectionwise Normal Spaces

A normal space is one in which any two disjoint closed sets can be separated by disjoint open sets. By induction, in a normal space any finite number of disjoint closed sets can be separated by disjoint open sets. Of course, the inductive reasoning cannot be carried over to the case of infinitely many disjoint closed sets. In the real line with the usual topology, the singleton sets \left\{x \right\}, where x is rational, are disjoint closed sets that cannot be simultaneously separated by disjoint open sets. In order to separate an infinite collection of disjoint closed sets, it makes sense to restrict on the type of collections of closed sets. A space X is collectionwise normal if every discrete collection of closed subsets of X can be separated by pairwise disjoint open subsets of X. The following is a more specific definition.

    Definition
    A space X is collectionwise normal if for every discrete collection \mathcal{A} of closed subsets of X, there exists a pairwise disjoint collection \mathcal{U}=\left\{U_A: A \in \mathcal{A} \right\} of open subsets of X such that A \subset U_A for each A \in \mathcal{A}.

For more details about the definitions of collectionwise normality, see “Definitions of Collectionwise Normal Spaces”. The implications displayed above are repeated below. None of the arrows is reversible.

    \text{ }
    \text{paracompact} \Longrightarrow \text{collectionwise normal} \Longrightarrow \text{normal}

As indicated above, Bing’s Example G is an example of a normal and not collectionwise normal space (see the blog post “Bing’s Example G”). The propositions in the next section can be used to obtain collectionwise normal spaces that are not paracompact.

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When Normal implies Collectionwise Normal

Being able to simultaneously separate any discrete collection of closed sets is stronger than the property of merely being able to separate finite collection of disjoint closed sets. It turns out that the stronger property of collectionwise normality is required only for separating uncountable discrete collections of closed sets. As the following lemma shows, normality is sufficient to separate any countable discrete collection of closed sets.

Lemma 1
Let X be a normal space. Then for every discrete collection \left\{C_1,C_2,C_3,\cdots \right\} of closed subsets of X, there exists a pairwise disjoint collection \left\{O_1,O_2,O_3,\cdots \right\} of open subsets of X such that C_i \subset O_i for each i.

Proof of Lemma 1
Let \left\{C_1,C_2,C_3,\cdots \right\} be a discrete collection of closed subsets of X. For each i, choose disjoint open sets U_i and V_i such that C_i \subset U_i and \cup \left\{C_j: j \ne i \right\} \subset V_i. Let O_1=U_1. For each i>1, let O_i=U_i \cap V_1 \cap \cdots \cap V_{i-1}. It follows that O_i \cap O_j = \varnothing for all i \ne j. It is also clear that for each i, C_i \subset O_i. \blacksquare

We have the following propositions.

Proposition 2
Let X be a normal space. If all discrete collections of closed subsets of X are at most countable, then X is collectionwise normal.

Proposition 3
Let X be a normal space. If all closed and discrete subsets of X are at most countable (such a space is said to have countable extent), then X is collectionwise normal.

Proposition 4
Any normal and countably compact space is collectionwise normal.

Proposition 2 follows from Lemma 1. As noted in Proposition 3, any space in which all closed and discrete subsets are countable is said to have countable extent. It is easy to verify that X has countable extent if and only if all discrete collections of closed subsets of X are at most countable. If X is a countably compact space, then every infinite subset of X has a limit point. Thus Proposition 4 follows from the fact that any countably compact space has countable extent.

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Paracompact Spaces

One way to find a collectionwise normal space that is not paracompact is to find a non-paracompact space that satisfies Propositions 3, 4 or 5. For example, \omega_1, the space of all countable ordinals with the order topology, is not paracompact. Since \omega_1 is normal and countably compact, it is collectionwise normal by Proposition 4. For a basic discussion of \omega_1 as a topological space, see “The First Uncountable Ordinal”.

As the following theorem shows, paracompact spaces are collectionwise normal. Thus the class of collectionwise normal spaces includes all metric spaces and paracompact spaces.

Theorem 5
If a space X is paracompact, then X is collectionwise normal.

Proof of Theorem 5
Let X be a paracompact space. Let \mathcal{A} be a discrete collection of closed subsets of X. For each x \in A, let O_x be open such that x \in O_x and O_x meets at most one element of \mathcal{A}. Let \mathcal{O}=\left\{O_x: x \in X \right\}. By the paracompactness of X, \mathcal{O} has a locally finite open refinement \mathcal{V}=\left\{V_x: x \in X \right\} such that V_x \subset O_x for each x \in X.

For each A \in \mathcal{A}, let W_A=\cup \left\{\overline{V}: V \in \mathcal{V} \text{ and } \overline{V} \cap A=\varnothing \right\} and let U_A=X-W_A. Each W_A is a closed set since \mathcal{V} is locally finite. Thus each U_A is open. Furthermore, for each A \in \mathcal{A}, A \subset U_A. It is easily checked that \left\{U_A: A \in \mathcal{A} \right\} is pairwise disjoint. \blacksquare

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Reference

  1. Bing, R. H., Metrization of Topological Spaces, Canad. J. Math., 3, 175-186, 1951.
  2. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  3. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

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\copyright \ \ 2012

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