Pair of spaces

In mathematics, particularly algebraic topology, a pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace (with the subspace topology).

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.

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.