Topological pair

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

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 ${\displaystyle A}$ are made equivalent to 0, when considered as chains in ${\displaystyle X}$.

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

There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space ${\displaystyle X}$ to the pair ${\displaystyle (X,\varnothing )}$.

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 x₀, one writes the triple as (X, A, B, x₀), where x₀ ∈ B ⊂ A ⊂ X.^[1]

References

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