In mathematics, more specifically algebraic topology, a pair
is shorthand for an inclusion of topological spaces
is assumed to be a cofibration.
A pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace.
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
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 x0, one writes the triple as (X, A, B, x0), where x0 ∈ B ⊂ A ⊂ X.
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