In mathematics, more specifically algebraic topology, a pair ( X , A ) {\displaystyle (X,A)} is shorthand for an inclusion of topological spaces i : A ↪ X {\displaystyle i\colon A\hookrightarrow X} . Sometimes i {\displaystyle i} is assumed to be a cofibration. A morphism from ( X , A ) {\displaystyle (X,A)} to ( X ′ , A ′ ) {\displaystyle (X',A')} is given by two maps f : X → X ′ {\displaystyle f\colon X\rightarrow X'} and
g : A → A ′ {\displaystyle g\colon A\rightarrow A'} such that i ′ ∘ g = f ∘ i {\displaystyle i'\circ g=f\circ i} . 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 {\displaystyle A} are made equivalent to 0, when considered as chains in X {\displaystyle X} . Heuristically, one often thinks of a pair ( X , A ) {\displaystyle (X,A)} as being akin to the quotient space X / A {\displaystyle X/A} . There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space X {\displaystyle X} to the pair ( X , ∅ ) {\displaystyle (X,\varnothing )} . 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.
References
Patty, C. Wayne (2009), Foundations of Topology (2nd ed.), p. 276.
