Pair of topological space and subspace

Definition

A pair of topological space and subspace, sometimes simply called a pair or pair of spaces is the following pieces of data:

• A topological space.
• A subset of the topological space, viewed as a subspace under the subspace topology.

If the topological space is $X$ and the subset is $A$, the pair is denoted $(X,A)$.

Such pairs are objects in the category of pairs of topological space and subspace.

Particular cases

Pair where the subset is empty

The pair $(X,\{\})$ is a special case of a pair. Here, the subset is empty. Such pairs can be thought of as topological spaces in an ordinary sense. More precisely, there is a functor from the category of topological spaces with continuous maps to the category of pairs of topological space and subspace that sends a topological space to the pair of itself and the empty subset.

Pair where the subset has size one

A pair where the subset is a singleton subset can be identified with the based topological space with the same space and where the basepoint is chosen as the unique element of that singleton subset. In other words, the pair $(X,\{x_{0}\})$ can be identified naturally with the based topological space $(X,x_{0})$.