## Mathematics, Department of

## Date of this Version

2-1961

## Abstract

The examples usually given as instances of topological spaces that have *T*_{1}-separation but not *T*_{2}-separation (Hausdorff) also have the property that some compact subset is not closed. This with the classic result concerning closedness of compact subsets of a Hausdorff space suggests the question of the equivalence of Hausdorff separation and the condition that the class of compact subsets be a subclass of the class of the closed subsets of a given space. The following is a simple result of this type and may be of some use in an introductory course in point set topology.

THEOREM. *If X is a space satisfying the first axiom of countability, then a necessary and sufficient condition that X be a Hausdorff space is that the class of compact subsets of X be a subclass of the class of closed subsets of X.*

Only the sufficiency need be considered here.

## Comments

Published in

The American Mathematical Monthly,Vol. 68, No. 2 (Feb., 1961), p. 164 Copyright 1961 Mathematical Association of America Used by permission.