Category of pairs of topological space and subspace

This article describes a category (in the mathematical sense) whose objects are pair of topological space and subspaces, and whose morphisms are continuous map between pairs of topological space and subspaces. In other words, it gives a category structure to the collection of all pair of topological space and subspaces.
View other category structures on pair of topological space and subspaces

Definition

Aspect	Name	Definition/description
objects	pair of topological space and subspace	A pair $(X,A)$ is a topological space $X$ along with a subset $A$, where the subset is viewed as a subspace under the subspace topology.
morphisms	continuous map between pairs of topological space and subspace	A continuous map $f:(X,A)\to (Y,B)$ of pairs of topological spaces is a continuous map $f:X\to Y$ such that $f(A)\subseteq B$. Note that the restriction of $f$ gives a continuous map from $A$ to $B$.
composition law for morphisms	compose as set maps	Compose the maps as set maps on the whole space; the induced maps on the subspaces also get composed.