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

If a topological spaceMathworldPlanetmath is HausdorffPlanetmathPlanetmath (T2), 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 accessiblePlanetmathPlanetmath (T1). There are spaces that are T1 and have compact sets that are not closed, and there are spaces in which compact sets are always closed but that are not T2.

Let X be an infinite setMathworldPlanetmath with the finite complement topology. Singletons are finite, and therefore closed, so X is T1. Let SβŠ‚X, and let 𝔽 be an open cover of S. Let Fβˆˆπ”½. Then Xβˆ–F is finite. Choosing a member of 𝔽 for each remaining element of S shows that 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 numberMathworldPlanetmath n, let Un be all the elements of C except for all the Sk, where k>n. Then Un is open for each n, and {Un∣nβˆˆβ„•} covers C, but has no finite subset that covers C, contradicting the fact that C is compact. This contradictionMathworldPlanetmathPlanetmath arose by assuming a compact subset of Y was infinite, all compact subsets of Y are finite. Y is T1 (singleton sets are countableMathworldPlanetmath), 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