a compact set in a Hausdorff space is closed


Theorem. A compact set in a Hausdorff space is closed.

Proof. Let A be a compact set in a Hausdorff space X. The case when A is empty is trivial, so let us assume that A is non-empty. Using this theorem (http://planetmath.org/APointAndACompactSetInAHausdorffSpaceHaveDisjointOpenNeighborhoods), it follows that each point y in A has a neighborhoodMathworldPlanetmathPlanetmath Uy, which is disjoint to A. (Here, we denote the complement of A by A.) We can therefore write

A = yAUy.

Since an arbitrary union of open sets is open, it follows that A is closed.

Note. 
The above theorem can, for instance, be found in [1] (page 141), or [2] (SectionPlanetmathPlanetmath 2.1, Theorem 2).

References

  • 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
  • 2 I.M. Singer, J.A.Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer-Verlag, 1967.
Title a compact set in a Hausdorff space is closed
Canonical name ACompactSetInAHausdorffSpaceIsClosed
Date of creation 2013-03-22 13:34:31
Last modified on 2013-03-22 13:34:31
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Theorem
Classification msc 54D10
Classification msc 54D30
Related topic ClosedSubsetsOfACompactSetAreCompact