Collectionwise Hausdorff space

In mathematics, in the field of topology, a topological space <math>X</math> is said to be '''collectionwise Hausdorff''' if given any closed discrete subset of <math>X</math>, there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.<ref>{{cite journal |last1=Tall |first1=Franklin |title=The density topology |journal=Pacific Journal of Mathematics |date=1976 |volume=62 |issue=1 |pages=275–284 |doi=10.2140/pjm.1976.62.275}}Definition 3.4</ref>

Here a subset <math>S\subseteq X</math> being ''discrete'' has the usual meaning of being a discrete space with the subspace topology (i.e., all points of <math>S</math> are isolated in <math>S</math>).<ref group=nb>If <math>X</math> is T₁ space, <math>S\subseteq X</math> being closed and discrete is equivalent to the family of singletons <math>\{\{s\}:s\in S\}</math> being a ''discrete family'' of subsets of <math>X</math> (in the sense that every point of <math>X</math> has a neighborhood that meets at most one set in the family). If <math>X</math> is not T₁, the family of singletons being a discrete family is a weaker condition. For example, if <math>X=\{a,b\}</math> with the indiscrete topology, <math>S=\{a\}</math> is discrete but not closed, even though the corresponding family of singletons is a discrete family in <math>X</math>.</ref>

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

* Metrizable spaces are collectionwise normal and hence collectionwise Hausdorff.

==Remarks== {{reflist|group=nb}}

== References == {{Reflist}} {{refbegin}} {{refend}}

Category:Properties of topological spaces