In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces. Sometimes i is assumed to be a cofibration. A morphism from (X,A) to (X',A') is given by two maps and such that. A pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace. The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of X by A . Pairs of spaces occur centrally in relative homology, 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. A related concept is that of a triple (X, A, B) , with B ⊂ A ⊂ X . Triples are used in homotopy theory. Often, for a pointed space with basepoint at x0 , one writes the triple as (X, A, B, x0) , where x0 ∈ B ⊂ A ⊂ X .