Topological pair

In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces i\colon A\hookrightarrow X. Sometimes i is assumed to be a cofibration. A morphism from (X,A) to (X',A') is given by two maps f\colon X\rightarrow X' and g\colon A\rightarrow A' such that  i' \circ g =f \circ i .

Pairs come up mainly in homology theory and cohomology theory, where chains in A are made equivalent to 0, when considered as chains in X.

Heuristically, one often thinks of a pair (X,A) as being akin to the quotient space X/A.

There is a functor from spaces to pairs, which sends a space X to the pair (X,\varnothing).

References

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