Put simply, a subspace is analogous to a subset of a topological space. Subspaces have powerful applications in topology.

Definition

Let ${\displaystyle (X,{\mathcal {T}})}$ be a topological space, and let ${\displaystyle X_{1}}$ be a subset of ${\displaystyle X}$. Define the open sets as follows:

A set ${\displaystyle U_{1}\subseteq X_{1}}$ is open in ${\displaystyle X_{1}}$ if there exists a a set ${\displaystyle U\in {\mathcal {T}}}$ such that ${\displaystyle U_{1}=U\bigcap X_{1}}$

An important idea to note from the above definitions is that a set not being open or closed does not prevent it from being open or closed within a subspace. For example, ${\displaystyle (0,1)}$ as a subspace of itself is both open and closed.