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 $\tau$).
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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.
A topological subspace is frequently referred just as a subspace if it has been established what it is a subspace of.
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): subspace: 2. (of a topological space)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): topological space
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): subspace: 2. (of a topological space)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): topological space
- 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (next): $\S 3$: New Spaces From Old: Subspaces