Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space ${\displaystyle X}$ is said to be collectionwise Hausdorff if given any closed discrete subset of ${\displaystyle X}$, 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 ${\displaystyle S\subseteq X}$ being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of ${\displaystyle S}$ are isolated in ${\displaystyle S}$).^{[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 ${\displaystyle S}$ of ${\displaystyle X}$, every singleton ${\displaystyle \{s\}}$ ${\displaystyle (s\in S)}$ is closed in ${\displaystyle X}$ and the family of such singletons is a discrete family in ${\displaystyle X}$.)

• Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.

Remarks

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

References

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