Topological pair

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

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 ${\displaystyle (X,\varnothing )}$.

References

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