Definition:Topological Subspace

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

Let $\varnothing \subset H \subseteq A$ be a non-null subset of $T$.

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

The set $\vartheta_H$ is defined as $\vartheta_H = \left\{{U \cap H: U \in \vartheta}\right\}$, and is called the relative topology, the induced topology or the subspace topology on $H$.

The fact that $T_H = \left({H, \vartheta_H}\right)$ is a topological space is proved in Topological Subspace is a Topological Space.