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 ThePropertyThatCompactSetsInASpaceAreClosedLiesStrictlyBetweenT1AndT2 2013-03-22 17:38:59 2013-03-22 17:38:59 dfeuer (18434) dfeuer (18434) 5 dfeuer (18434) Result msc 54D30 msc 54D10 T1Space T2Space Compact