Definition:Topological Subspace/Also known as

Topological Subspace: 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.

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.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces