Compactly-supported homology
In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group Hn of every pair of spaces
is naturally isomorphic to the direct limit of the nth relative homology groups of pairs, where Y varies over compact subspaces of X and B varies over compact subspaces of A.
Singular homology is compactly supported, since each singular chain is a finite sum of simplices, which are compactly supported. Strong homology is not compactly supported.
If one has defined a homology theory over compact pairs, it is possible to extend it into a compactly supported homology theory in the wider category of Hausdorff pairs with A closed in X, by defining that the homology of a Hausdorff pair is the direct limit over pairs, where Y, B are compact, Y is a subset of X, and B is a subset of A.
