- Σ-compact space
In mathematics, a

topological space is said to be**σ-compact**if it is the union of countably many compact subspaces.A space is said to be

**σ-locally compact**if it is both σ-compact and locally compact.**Properties and Examples*** Every compact space is σ-compact.

* Moreover, every σ-compact space is Lindelöf (i.e. every

open cover has a countable subcover).* The reverse implications of the previous two examples do not hold. For example, standard

Euclidean space (**R**^{"n"}) is σ-compact but not compact, and thelower limit topology on the real line is Lindelöf but not σ-compact or compact. In fact, thecountable complement topology is Lindelof but neither σ-compact nor locally compact.* Let "X" be a Hausdorff,

Baire space that is also σ-compact. Then "X" must belocally compact at at least one point.* If "G" is a

topological group and "G" is locally compact at one point, then "G" is locally compact everywhere. Therefore, the previous property tells us that if "G" is a σ-compact topological group that is also a Baire space, then "G" is locally compact. This shows that for topological groups that are also Baire spaces, σ-compactness implies local compactness* We can conclude from the previous property that

**R**^{ω}is not σ-compact (if it were σ-compact, it would locally compact since**R**^{ω}is a topological group that is also a Baire space)**ee also***

Exhaustion by compact sets

*Locally compact space

*Lindelöf space

*Wikimedia Foundation.
2010.*

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