Compact polyhedral pair

Definition

A pair $(X,A)$ where $X$ is a topological space and $A$ is a subspace, is termed a compact polyhedral pair if there is a (finite) simplicial complex $K$ with a subcomplex $L$, and a triangulation (viz, a homeomorphism) $h:|K|\to X$ such that $h(|L|)=A$.

Complex polyhedral pairs are important because we can do homology theory for these instead of just for polyhedra.