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:
 * $\tau_H := \set {U \cap H: U \in \tau} \subseteq \powerset H$

where $\powerset H$ denotes the power set of $H$.

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 $H$ (induced by $\tau$).

Also known as
The subspace topology $\tau_H$ induced by $\tau$ can be referred to as just the induced topology (on $H$) if there is no ambiguity.

The term relative topology can also be found.

Also see

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