weakly countably compact
A topological space is said to be weakly countably compact (or limit point compact) if every infinite subset of has a limit point.
Every countably compact space is weakly countably compact. The converse is true in 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 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 |