topological pair

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

$g\backslash colon\; A\; \backslash rightarrow\; A\text{'}$ such that $i\text{'}\; \backslash circ\; g\; =f\; \backslash circ\; i$.

A pair of spaces is an ordered pair {{math|(X, A)}} where {{math|X}} is a topological space and {{math|A}} a subspace. The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of {{math|X}} by {{math|A}}. Pairs of spaces occur centrally in relative homology,{{cite book | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | url = http://pi.math.cornell.edu/~hatcher/AT/ATpage.html}} homology theory and cohomology theory, where chains in $A$ are made equivalent to 0, when considered as chains in $X$.

Heuristically, one often thinks of a pair $(X,A)$ as being akin to the quotient space $X/A$.

There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space $X$ to the pair $(X,\; \backslash varnothing)$.

A related concept is that of a triple {{math|(X, A, B)}}, with {{math|B ⊂ A ⊂ X}}. Triples are used in homotopy theory. Often, for a pointed space with basepoint at {{math|x₀}}, one writes the triple as {{math|(X, A, B, x₀)}}, where {{math|x₀ ∈ B ⊂ A ⊂ X}}.