| 释义 |
- Examples
- Properties
- See also
- References
In mathematics a topological space is countably compact if every countable open cover has a finite subcover. Examples- The first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.
Properties- Every compact space is countably compact.
- A countably compact space is compact if and only if it is Lindelöf.
- A countably compact space is always limit point compact.
- For T1 spaces, countable compactness and limit point compactness are equivalent.
- For metrizable spaces, countable compactness, sequential compactness, limit point compactness and compactness are all equivalent.
- The example of the set of all real numbers with the standard topology shows that neither local compactness nor σ-compactness nor paracompactness imply countable compactness.
See also- Sequentially compact space
- Compact space
- Limit point compact
References
| author = James Munkres
| year = 1999
| title = Topology
| edition = 2nd
| publisher = Prentice Hall
| isbn = 0-13-181629-2
}} 1 : Properties of topological spaces |