weakly countably compact

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

Every countably compact space is weakly countably compact. The converse is true in $\mathrm{T}_{1}$ spaces (http://planetmath.org/T1Space).

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

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