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 T1 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 |