Definition:Topological Subspace

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

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

Define:
 * $\tau_H := \left\{{U \cap H: U \in \tau}\right\} \subseteq \mathcal P \left({H}\right)$

Then the topological space $T_H = \left({H, \tau_H}\right)$ 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 $T_H = \left({H, \tau_H}\right)$ is a topological space.