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 $i'\circ g=f\circ i$ .

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 $A$ 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 the category of topological spaces to the category of pairs of spaces, which sends a space $X$ to the pair $(X,\varnothing )$ .

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.

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

[hatcher-1] Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.

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