From: dontmailme@iname.com (Steamer)
Subject: Re: Q: when are compact sets closed?
Date: Thu, 29 Apr 1999 17:15:42 GMT
Newsgroups: sci.math
Keywords: KC spaces

Volker W. Elling asked:

> In which class of topological spaces are compact sets necessarily
> closed?
>
> At least in all Hausdorff spaces, but not in all $T_1$ spaces. Does
> anybody
> know a reference or an answer?

Spaces in which all compact sets are closed are sometimes called
KC spaces. I don't think there is any neat (and non-obvious)
characterization of KC spaces, but there is one for compact KC
spaces:

  A compact space is KC if and only if it is maximal compact.

This is straightforward to prove using Alexander's Subbase Theorem.

KC is a strictly weaker property than T_2. For example, the
1-point compactification of the rationals is KC but not T_2.
(The 1-point compactification of X is KC if and only if X is a
KC k-space.)

Define a US space to be a topological space in which no sequence has
more than one limit. Then every KC space is US (and every US space
is obviously T_1). There are US spaces that are not KC, but I can't
think of one offhand. If my memory serves me correctly, there is
one in Albert Wilansky's paper "Between T_1 and T_2" (AMM 74 (1967)
261-266).

Wilansky's book "Topology for Analysis" (Ginn, 1970) has a few things
on KC spaces.

S.