Topological pair

In mathematics, more specifically algebraic topology, a pair ( X , A ) is shorthand for an inclusion of topological spaces i : A ↪ X . Sometimes i is assumed to be a cofibration. A morphism from ( X , A ) to ( X ′ , A ′ ) is given by two maps f : X → X ′ and g : A → A ′ such that i ′ ∘ g = f ∘ i .

Pairs come up mainly in 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 spaces to pairs, which sends a space X to the pair ( X , ∅ ) .