countable complement topology

Let X be an infinite setMathworldPlanetmath. We define the countable complement topology on X by declaring the empty setMathworldPlanetmath to be open, and a non-empty subset UX to be open if X\U is countableMathworldPlanetmath.

If X is countable, then the countable complement topology is just the discrete topology, as the complement of any set is countable and thus open.

Though defined similarly to the finite complement topologyMathworldPlanetmathPlanetmath, the countable complement topology lacks many of the strong compactness properties of the finite complement topology. For example, the countable complement topology on an uncountable set gives an example of a topological space that is not weakly countably compact (but is pseudocompact).

Title countable complement topology
Canonical name CountableComplementTopology
Date of creation 2013-03-22 14:37:56
Last modified on 2013-03-22 14:37:56
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 5
Author mathcam (2727)
Entry type Definition
Classification msc 54A05
Synonym cocountable topology