Pair of spaces

In mathematics, particularly algebraic topology, 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]

A related concept is that of a triple $(X, A, B)$ , with $B \subset A \subset X$ . 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 $x 0 \in B \subset A \subset X$ . ^[1]

References

^ ^a ^b Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0. http://www.math.cornell.edu/~hatcher/AT/ATpage.html.