Σ-compact space

• Every compact space is σ-compact, and every σ-compact space is Lindelöf (i.e. every open cover has a countable subcover).Steen, p. 19. The reverse implications do not hold, for example, standard Euclidean space (Rⁿ) is σ-compact but not compact,Steen, p. 56. and the lower limit topology on the real line is Lindelöf but not σ-compact.Steen, p. 75–76. In fact, the countable complement topology on any uncountable set is Lindelöf but neither σ-compact nor locally compact.Steen, p. 50. However, it is true that any locally compact Lindelöf space is σ-compact.
• (The irrational numbers) $\backslash mathbb\; R\backslash setminus\backslash mathbb\; Q$ is not σ-compact.{{cite book |last1=Hart |first1=K.P. |last2=Nagata |first2=J. |last3=Vaughan |first3=J.E. |title=Encyclopedia of General Topology |date=2004 |publisher=Elsevier |isbn=0-444-50355-2 |page=170}}
• 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 R^ω is not σ-compact: if it were σ-compact, it would necessarily be locally compact since R^ω is a topological group that is also a Baire space.
• Every hemicompact space is σ-compact.Willard, p. 126. The converse, however, is not true;Willard, p. 126. 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.Willard, p. 126.
• A σ-compact space X is second category (respectively Baire) if and only if the set of points at which is X is locally compact is nonempty (respectively dense) in X.Willard, p. 188.

• {{annotated link|Exhaustion by compact sets}}
• {{annotated link|Lindelöf space}}
• {{annotated link|Locally compact space}}

{{reflist}}

• 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}}

Category:Compactness (mathematics)

Category:General topology

Category:Properties of topological spaces