| 释义 |
- Properties and examples
- See also
- Notes
- References
{{lowercase}}In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces.[1] A space is said to be σ-locally compact if it is both σ-compact and locally compact.[2] Properties and examples- Every compact space is σ-compact, and every σ-compact space is Lindelöf (i.e. every open cover has a countable subcover).[3] The reverse implications do not hold, for example, standard Euclidean space (Rn) is σ-compact but not compact,[4] and the lower limit topology on the real line is Lindelöf but not σ-compact.[5] In fact, the countable complement topology is Lindelöf but neither σ-compact nor locally compact.[6]
- A Hausdorff, Baire space that is also σ-compact, 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, Hausdorff topological group that is also a Baire space, then G is locally compact. This shows that for Hausdorff topological groups that are also Baire spaces, σ-compactness implies local compactness.
- The previous property implies for instance that Rω is not σ-compact: if it were σ-compact, it would necessarily be locally compact since Rω is a topological group that is also a Baire space.
- Every hemicompact space is σ-compact.[7] The converse, however, is not true;[8] for example, the space of rationals, with the usual topology, is σ-compact but not hemicompact.
- The product of a finite number of σ-compact spaces is σ-compact. However the product of an infinite number of σ-compact spaces may fail to be σ-compact.[9]
- A σ-compact space X is second category (resp. Baire) if and only if the set of points at which is X is locally compact is nonempty (resp. dense) in X.[10]
See also- Exhaustion by compact sets
- Lindelöf space
Notes1. ^Steen, p.19; Willard, p. 126.
2. ^Steen, p. 21.
3. ^Steen, p. 19.
4. ^Steen, p. 56.
5. ^Steen, p. 75–76.
6. ^Steen, p. 50.
7. ^Willard, p. 126.
8. ^Willard, p. 126.
9. ^Willard, p. 126.
10. ^Willard, p. 188.
References- Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). {{ISBN|0-03-079485-4}}.
- {{cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | isbn=0-486-43479-6}}
{{DEFAULTSORT:Compact Space}} 3 : Compactness (mathematics)|General topology|Properties of topological spaces |