- Σ-compact space
In mathematics, a
topological spaceis 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 coverhas 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 topologyon the real line is Lindelöf but not σ-compact or compact. In fact, the countable complement topologyis Lindelof but neither σ-compact nor locally compact.
* Let "X" be a Hausdorff,
Baire spacethat is also σ-compact. Then "X" must be locally compactat at least one point.
* If "G" is a
topological groupand "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)
Exhaustion by compact sets
Locally compact space
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