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. spaces). Let be a space. Let be a collection of subsets of . We say is pairwise disjoint if for all where . We say is a discrete collection of subsets of if for each , there is an open neighborhood with such that meets at most one element of . When is such a collection, we also say that is discrete in (or discrete if is understood).

**Definition 1**A space is said to be collectionwise normal if for every discrete collection of closed subsets of , there is a collection of open subsets of such that is pairwise disjoint and for each .

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 is said to be collectionwise normal if for every discrete collection of subsets of , there is a collection of open subsets of such that is pairwise disjoint and for each .

It is clear that Definition 2 implies Definition 1. For any discrete collection of subsets, is also a discrete collection. Thus Definition 1 implies Definition 2. The following is another way of defining collectionwise normal.

**Definition 3**A space is said to be collectionwise normal if for every discrete collection of closed subsets of , there is a collection of open subsets of such that is a discrete collection and for each .

Since discrete collection is pairwise disjoint collection of sets, Definition 3 implies Definition 1. We show that Definition 1 implies Definition 3. Let be a collectionwise normal space according to Definition 1. Let be a discrete collection of closed subsets of . Let be as in Definition 1. Let and . Note that and are disjoint closed subsets of .

Suppose . Then the sets in are both open and closed. Thus is the desired discrete collection of open sets separating sets in . So assume that . Since is a normal space, we have and , disjoint open subsets of , such that and . For each , let . Let .

We claim that is a discrete collection of open sets separating sets in . Let . Suppose . We have for some . Then meets only one set of , namely . So assume . Then and does not meet any . Thus is a discrete collection of open sets. It is clear that for each .

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**

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

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