# Definition:Topological Subspace

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## Contents

## 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 $H$.

## Also known as

The **subspace topology** $\tau_H$ is also known as the **relative topology** or the **induced topology** on $H$.

## Also see

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

- Results about
**topological subspaces**can be found here.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.4$: Subspaces: Definition $3.4.1$