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 (*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,^{[1]} 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 the category of topological spaces to the category of pairs of spaces, which sends a space to the pair .

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 *x*_{0}, one writes the triple as (*X*, *A*, *B*, *x*_{0}), where *x*_{0} ∈ *B* ⊂ *A* ⊂ *X*.^{[1]}

## References

- ^
^{a}^{b}Hatcher, Allen (2002).*Algebraic Topology*. Cambridge University Press. ISBN 0-521-79540-0.

- Patty, C. Wayne (2009),
*Foundations of Topology*(2nd ed.), p. 276.