weakly countably compact

A topological spaceMathworldPlanetmath X is said to be weakly countably compact (or limit point compact) if every infinite subset of X has a limit pointPlanetmathPlanetmath.

Every countably compact space is weakly countably compact. The converseMathworldPlanetmath is true in T1 spaces (http://planetmath.org/T1Space).

A metric space is weakly countably compact if and only if it is compactPlanetmathPlanetmath.

An easy example of a space X that is not weakly countably compact is any infinite set with the discrete topology. A more interesting example is the countable complement topology on an uncountable set.

Title weakly countably compact
Canonical name WeaklyCountablyCompact
Date of creation 2013-03-22 12:06:46
Last modified on 2013-03-22 12:06:46
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Definition
Classification msc 54D30
Synonym limit point compact
Synonym limit-point compact
Related topic Compact
Related topic CountablyCompact
Related topic SequentiallyCompact
Related topic PseudocompactSpace
Defines limit point compactness
Defines weak countable compactness