Definition:Topological Subspace

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

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


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

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

  • Results about topological subspaces can be found here.