The property that compact sets in a space are closed lies strictly between T1 and T2

If a topological space is Hausdorff ( $T_{2}$ ), then every compact subset of that space is closed. If every compact subset of a space is closed, then (since singletons are always compact) then the space is accessible ( $T_{1}$ ). There are spaces that are $T_{1}$ and have compact sets that are not closed, and there are spaces in which compact sets are always closed but that are not $T_{2}$ .

Let $X$ be an infinite set with the finite complement topology. Singletons are finite, and therefore closed, so $X$ is $T_{1}$ . Let $S\subset X$ , and let $\mathbb{F}$ be an open cover of $S$ . Let $F\in\mathbb{F}$ . Then $X\setminus F$ is finite. Choosing a member of $\mathbb{F}$ for each remaining element of $S$ shows that $\mathbb{F}$ has a finite subcover. Thus, every subset of $X$ is compact. An infinite subset of $X$ will then be compact, but not closed.

Let $Y$ be an uncountable set with the countable complement topology. No two open sets are disjoint, so $Y$ is not Hausdorff. Let $C$ be a compact subset of $Y$ . I shall show that $C$ is finite. Suppose $C$ is infinite, and let $S$ be an infinite sequence in $C$ without any repetitions. For any natural number $n$ , let $U_{n}$ be all the elements of $C$ except for all the $S_{k}$ , where $k>n$ . Then $U_{n}$ is open for each $n$ , and $\{U_{n}\mid n\in\mathbb{N}\}$ covers $C$ , but has no finite subset that covers $C$ , contradicting the fact that $C$ is compact. This contradiction arose by assuming a compact subset of $Y$ was infinite, all compact subsets of $Y$ are finite. $Y$ is $T_{1}$ (singleton sets are countable), so all compact subsets of $Y$ are closed.

These examples were suggested by the person known as Polytope on EFNet’s math channel.

Title	The property that compact sets in a space are closed lies strictly between T1 and T2
Canonical name	ThePropertyThatCompactSetsInASpaceAreClosedLiesStrictlyBetweenT1AndT2
Date of creation	2013-03-22 17:38:59
Last modified on	2013-03-22 17:38:59
Owner	dfeuer (18434)
Last modified by	dfeuer (18434)
Numerical id	5
Author	dfeuer (18434)
Entry type	Result
Classification	msc 54D30
Classification	msc 54D10
Related topic	T1Space
Related topic	T2Space
Related topic	Compact