Collectionwise Hausdorff space explained

X is said to be collectionwise Hausdorff if given any closed discrete subset of X , there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.^[1]

Here a subset

S\subseteq X being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of S are isolated in S ).^[2]

Properties

• Every T₁ space that is collectionwise Hausdorff is also Hausdorff.
• Every collectionwise normal space is collectionwise Hausdorff. (This follows from the fact that given a closed discrete subset

S of X , every singleton \{s\} (s\in S) is closed in X and the family of such singletons is a discrete family in X .)

• Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.

Notes and References

1. FD Tall, The density topology, Pacific Journal of Mathematics, 1976
2. If X is T₁ space, S\subseteq X being closed and discrete is equivalent to the family of singletons \{\{s\}:s\in S\} being a discrete family of subsets of X (in the sense that every point of X has a neighborhood that meets at most one set in the family). If X is not T₁, the family of singletons being a discrete family is a weaker condition. For example, if X = \{a,b\} with the indiscrete topology, S = \{a\} is discrete but not closed, even though the corresponding family of singletons is a discrete family in X .