mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 topolog ...

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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

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 topological pair, pairs of spaces. The relative homology is useful and important in sev ...

group H_''n''(''X'', ''A'') of every

pair of spaces In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces i\colon A \hookrightarrow X. Sometimes i is assumed to be a cofibration. A morphism from (X,A) to (X',A') is given by two maps ...

:(''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 natur ...

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 categor ...

of the ''n''th relative homology groups of pairs (''Y'', ''B''), where ''Y'' varies over compact subspaces 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''-d ...

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 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