Definition:Topological Subspace

Definition
Let $T = \struct{S, \tau}$ be a topological space.

Let $H \subseteq S$ be a non-empty subset of $S$.

Define:
 * $\tau_H := \set{U \cap H: U \in \tau} \subseteq \mathcal P \paren{H}$

Then the topological space $T_H = \struct{H, \tau_H}$ is called a (topological) subspace of $T$.

The set $\tau_H$ is referred to as the subspace topology on $H$.

Also known as
The subspace topology $\tau_H$ is also known as the relative topology or the induced topology on $H$.

Also see

 * Topological Subspace is Topological Space which proves that $T_H = \struct{H, \tau_H}$ is a topological space.