Topology/Subspaces

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

Definition

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

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

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, $(0,1)$ as a subspace of itself is both open and closed.