## You are here

Homeweakly countably compact

## Primary tabs

# 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.

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.

Defines:

limit point compactness, weak countable compactness

Keywords:

topology

Related:

Compact, CountablyCompact, SequentiallyCompact, PseudocompactSpace

Synonym:

limit point compact, limit-point compact

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

54D30*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff