In
mathematics, a
homology theory
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...
in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
is compactly supported if, in every degree ''n'', the
relative homology group H
''n''(''X'', ''A'') of every
pair of spaces
:(''X'', ''A'')
is
naturally isomorphic to the
direct limit of the ''n''th relative homology groups of pairs (''Y'', ''B''), where ''Y'' varies over
compact subspace
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e ...
s of ''X'' and ''B'' varies over compact subspaces of ''A''.
[.]
Singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
is compactly supported, since each singular chain is a finite sum of
simplices, which are compactly supported.
Strong homology
Strong may refer to:
Education
* The Strong, an educational institution in Rochester, New York, United States
* Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas
* Strong School, New Haven, Connecticut, United St ...
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''.
References
{{topology-stub
Homology theory