# Definition:Topological Subspace

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## 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 $H$ (induced by $\tau$)**.

## Also known as

The **subspace topology $\tau_H$ induced by $\tau$** can be referred to as just the **induced topology (on $H$)** if there is no ambiguity.

The term **relative topology** can also be found.

## 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

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.4$: Subspaces: Definition $3.4.1$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction