 %CE%A3-compact Space
Get %CE%A3-compact Space essential facts below. View Videos or join the %CE%A3-compact Space discussion. Add %CE%A3-compact Space to your PopFlock.com topic list for future reference or share this resource on social media.
%CE%A3-compact Space

In mathematics, a topological space is said to be ?-compact if it is the union of countably many compact subspaces.

A space is said to be ?-locally compact if it is both ?-compact and locally compact.

## Properties and examples

• Every compact space is ?-compact, and every ?-compact space is Lindelöf (i.e. every open cover has a countable subcover). The reverse implications do not hold, for example, standard Euclidean space (Rn) is ?-compact but not compact, and the lower limit topology on the real line is Lindelöf but not ?-compact. In fact, the countable complement topology on any uncountable set is Lindelöf but neither ?-compact nor locally compact. However, it is true that any locally compact Lindelöf space is ?-compact.
• A Hausdorff, Baire space that is also ?-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 ?-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, ?-compactness implies local compactness.
• The previous property implies for instance that R? is not ?-compact: if it were ?-compact, it would necessarily be locally compact since R? is a topological group that is also a Baire space.
• Every hemicompact space is ?-compact. The converse, however, is not true; for example, the space of rationals, with the usual topology, is ?-compact but not hemicompact.
• The product of a finite number of ?-compact spaces is ?-compact. However the product of an infinite number of ?-compact spaces may fail to be ?-compact.
• A ?-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.