In mathematics, a topological space is said to be limit point compact or weakly countably compact has a limit point This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.

Properties and examples

• In a topological space, subsets without limit point are exactly those that are closed and discrete in the subspace topology. So a space is limit point compact if and only if all its closed discrete subsets are finite.

• is has a countably infinite closed discrete subspace.

• ; (2) an infinite set with the discrete topology; (3) the countable complement topology on an uncountable set.

• Every countably compact space (and hence every compact space) is limit point compact.

• For T1 spacess, limit point compactness is equivalent to countable compactness.

• is the set of all integers with the discrete topology has the indiscrete topology is homeomorphic to the odd-even topology. This space is not T0. It is limit point compact because every nonempty subset has a limit point.

• An example of T0 the set of all real numbers, with the right order topology

• For metrizable spaces, compactness, countable compactness, limit point compactness, and sequential compactness are all equivalent.

• Closed subspaces of a limit point compact space are limit point compact.

• is not limit point compact.

• A limit point compact space need not be pseudocompact

• A limit point compact space need not be compact. The integers serve as an example of this fact.

• A pseudocompact space need not be limit point compact. An example is given by an uncountable set with the cocountable topology.

• Every normal pseudocompact space is limit point compact.Proof By the Tietze extension theorem is not pseudocompact.

• Limit point compact spaces have countable extent.

See also

Notes

References