Topological pair
In mathematics, more specifically algebraic topology, a pair is shorthand for an inclusion of topological spaces. Sometimes is assumed to be a cofibration. A morphism from to is given by two maps and
such that.
A pair of spaces is an ordered pair where is a topological space and a subspace. The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of by. Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains in are made equivalent to 0, when considered as chains in.
Heuristically, one often thinks of a pair as being akin to the quotient space.
There is a functor from spaces to pairs, which sends a space to the pair.
A related concept is that of a triple, with. Triples are used in homotopy theory. Often, for a pointed space with basepoint at, one writes the triple as, where.
