Topological pair
In mathematics, more specifically algebraic topology, a pair formula_1 is shorthand for an inclusion of topological spaces formula_2. Sometimes formula_3 is assumed to be a cofibration. A morphism from formula_1 to formula_5 is given by two maps formula_6 and formula_7 such that formula_8. A pair of spaces is an ordered pair where is a topological space and 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 by . Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains  in formula_9 are made equivalent to 0, when considered as chains in formula_10. Heuristically, one often thinks of a pair formula_1 as being akin to the quotient space formula_12. There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space formula_10 to the pair formula_14. 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 .