Definition:Topological Subspace

Definition

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

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

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

The set $\tau_H$ is defined as:

$\tau_H := \left\{{U \cap H: U \in \tau}\right\}$

and is called the relative topology, the induced topology or the subspace topology on $H$.

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

Also see

• Results about topological subspaces can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$