mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a

homology theory In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...

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 pairs of spaces. The relative homology is useful and important in several ways. Intu ...

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 cate ...

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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...

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-dimensional ...

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