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 ...

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 In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...

group H_''n''(''X'', ''A'') of every pair of spaces :(''X'', ''A'') is

naturally isomorphic In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natu ...

to the

direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cat ...

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 In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

, 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