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.