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.