Compactly supported homology
{{Short description|Homology Theory in Algebraic Topology is Compactly Supported}}In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group Hn(X, A) of every pair of spaces
:(X, A)
is naturally isomorphic to the direct limit of the nth relative homology groups of pairs (Y, B), where Y varies over compact subspaces of X and B varies over compact subspaces of A.{{citation|title=Differential Algebraic Topology: From Stratifolds to Exotic Spheres|volume=110|series=Graduate Studies in Mathematics|first=Matthias|last=Kreck|publisher=American Mathematical Society|year=2010|isbn=9780821848982|page=95|url=https://books.google.com/books?id=iYyDAwAAQBAJ&pg=PA95}}.
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 (X, A) with A closed in X, by defining that the homology of a Hausdorff pair (X, A) is the direct limit over pairs (Y, B), where Y, B are compact, Y is a subset of X, and B is a subset of A.