Definitions of Collectionwise Normal Spaces

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The notion of collectionwise normal spaces is a property stronger than normal spaces. There are several ways of defining the notion of collectionwise normality. We show that they are equivalent.

We only consider spaces in which any set with only one point is considered a closed set (i.e. T_1 spaces). Let X be a space. Let \mathcal{A} be a collection of subsets of X. We say \mathcal{A} is pairwise disjoint if A \cap B = \varnothing for all A,B \in \mathcal{A} where A \ne B. We say \mathcal{A} is a discrete collection of subsets of X if for each x \in X, there is an open neighborhood O with x \in O such that O meets at most one element of \mathcal{A}. When \mathcal{A} is such a collection, we also say that \mathcal{A} is discrete in X (or discrete if X is understood).

    Definition 1
    A space X is said to be collectionwise normal if for every discrete collection \mathcal{A} of closed subsets of X, there is a collection \mathcal{U}=\left\{U_A: A \in \mathcal{A} \right\} of open subsets of X such that \mathcal{U} is pairwise disjoint and A \subset U_A for each A \in \mathcal{A}.

In other words, every discrete collection of closed sets can be separated by pairwise disjoint open sets. Clearly any collectionwise normal space is normal. When discrete collection of closed sets in the definition of “collectionwise normal” is replaced by discrete collection of singleton sets, the space is said to be collectionwise Hausdorff. Clearly any collectionwise normal space is collectionwise Hausdorff.

Some authors define a collectionwise normal space as one in which every discrete collection of sets can be separated by pairwise disjoint open sets. We have the following definition.

    Definition 2
    A space X is said to be collectionwise normal if for every discrete collection \mathcal{A} of subsets of X, there is a collection \mathcal{U}=\left\{U_A: A \in \mathcal{A} \right\} of open subsets of X such that \mathcal{U} is pairwise disjoint and A \subset U_A for each A \in \mathcal{A}.

It is clear that Definition 2 implies Definition 1. For any discrete collection \mathcal{A} of subsets, \mathcal{A}^*=\left\{\overline{A}: A \in \mathcal{A} \right\} is also a discrete collection. Thus Definition 1 implies Definition 2. The following is another way of defining collectionwise normal.

    Definition 3
    A space X is said to be collectionwise normal if for every discrete collection \mathcal{A} of closed subsets of X, there is a collection \mathcal{U}=\left\{U_A: A \in \mathcal{A} \right\} of open subsets of X such that \mathcal{U} is a discrete collection and A \subset U_A for each A \in \mathcal{A}.

Since discrete collection is pairwise disjoint collection of sets, Definition 3 implies Definition 1. We show that Definition 1 implies Definition 3. Let X be a collectionwise normal space according to Definition 1. Let \mathcal{A} be a discrete collection of closed subsets of X. Let \mathcal{U}=\left\{U_A: A \in \mathcal{A} \right\} be as in Definition 1. Let H=X-\cup \mathcal{U} and K=\cup \mathcal{A}. Note that H and K are disjoint closed subsets of X.

Suppose H=\varnothing. Then the sets in \mathcal{U} are both open and closed. Thus \mathcal{U} is the desired discrete collection of open sets separating sets in \mathcal{A}. So assume that H \ne \varnothing. Since X is a normal space, we have O_H and O_K, disjoint open subsets of X, such that H \subset O_H and K \subset O_K. For each A \in \mathcal{A}, let V_A=U_A \cap O_K. Let \mathcal{V}=\left\{V_A: A \in \mathcal{A} \right\}.

We claim that \mathcal{V} is a discrete collection of open sets separating sets in \mathcal{A}. Let x \in X. Suppose x \notin H. We have x \in U_A for some A \in \mathcal{A}. Then U_A meets only one set of \mathcal{A}, namely V_A. So assume x \in H. Then x \in O_H and O_H does not meet any V_A. Thus \mathcal{A} is a discrete collection of open sets. It is clear that A \subset V_A for each A \in \mathcal{A}.

Thus all three statements defining the notion of collectionwise normal spaces are equivalent. It is clear that any space satisfying any one of these collectionwise normal definitions is a normal space. The notion of collectionwise normality is stronger than normality. Bing’s Example G is a normal space that is not collectionwise normal (see “Bing’s Example G”).

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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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