Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space $X$ 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 $S$ ). [nb 1]

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.

Remarks

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 $X$ .

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References

FD Tall, The density topology, Pacific Journal of Mathematics, 1976

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[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 $X$ .

[1] FD Tall, The density topology, Pacific Journal of Mathematics, 1976