cofinite and cocountable topologies

The cofinite topologyMathworldPlanetmath on a set X is defined to be the topologyMathworldPlanetmath 𝒯 where

𝒯={AXXA is finite, or A=}.

In other words, the closed setsPlanetmathPlanetmath in the cofinite topology are X and the finite subsets of X.

Analogously, the cocountable topology on X is defined to be the topology in which the closed sets are X and the countableMathworldPlanetmath subsets of X.

The cofinite topology on X is the coarsest T1 topology ( on X.

The cofinite topology on a finite setMathworldPlanetmath X is the discrete topology. Similarly, the cocountable topology on a countable set X is the discrete topology.

A set X together with the cofinite topology forms a compactPlanetmathPlanetmath topological space.

Title cofinite and cocountable topologies
Canonical name CofiniteAndCocountableTopologies
Date of creation 2013-03-22 13:03:30
Last modified on 2013-03-22 13:03:30
Owner yark (2760)
Last modified by yark (2760)
Numerical id 21
Author yark (2760)
Entry type Definition
Classification msc 54B99
Related topic FiniteComplementTopology
Defines cofinite topology
Defines cocountable topology
Defines cofinite
Defines cocountable