Topological pair

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

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 [math]\displaystyle{ A }[/math] are made equivalent to 0, when considered as chains in [math]\displaystyle{ X }[/math].

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

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

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.

References

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