finite complement topology

Let X be a set. We can define the finite complementPlanetmathPlanetmath topologyMathworldPlanetmath on X by declaring a subset UX to be open if X\U is finite, or if U is all of X or the empty setMathworldPlanetmath. Note that this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to defining a topology by defining the closed setsPlanetmathPlanetmath in X to be all finite setsMathworldPlanetmath (and X itself).

If X is finite, the finite complement topology on X is clearly the discrete topology, as the complement of any subset is finite.

If X is countably infiniteMathworldPlanetmath (or larger), the finite complement topology gives a standard example of a space that is not HausdorffPlanetmathPlanetmath (each open set must contain all but finitely many points, so any two open sets must intersect).

In general, the finite complement topology on an infinite setMathworldPlanetmath satisfies strong compactness conditions (compactPlanetmathPlanetmath, σ-compact (, sequentially compact, etc.) since each open set in a cover contains ”almost all” of the points of X. On the other hand, the finite complement topology fails all but the simplest of separation axiomsMathworldPlanetmath since, as above, X is hyperconnected under this topology.

The finite complement topology is the coarsest T1-topology on a given set.

Title finite complement topology
Canonical name FiniteComplementTopology
Date of creation 2013-03-22 14:37:54
Last modified on 2013-03-22 14:37:54
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 54A05
Synonym cofinite topologyMathworldPlanetmath
Related topic CofiniteTopology
Related topic CofiniteAndCocountableTopology