Σ-compact space


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 the lower limit topology on the real line is Lindelöf but not σ-compact or compact. In fact, the countable 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 be locally 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 Rω is not σ-compact (if it were σ-compact, it would locally compact since Rω is a topological group that is also a Baire space)

ee also

*Exhaustion by compact sets
*Locally compact space
*Lindelöf space


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