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, 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 $B ⊂ A ⊂ X$. 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 $x_{0} ∈ B ⊂ A ⊂ X$.