# Topology/Subspaces

 Topology ← Sequences Subspaces Order →

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.

 Topology ← Sequences Subspaces Order →