Definition:Weakly Hereditary Property

Definition

Let $\xi$ be a property whose domain is the set of all topological spaces.

Then $\xi$ is a weakly hereditary property if and only if:

$\map \xi X \implies \map \xi Y$

where $Y$ is any closed set of $X$ when considered as a subspace.

That is, whenever a topological space has $\xi$, then so does any closed subspace.

Also see

• Definition:Hereditary Property (Topology)

Sources

• 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
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hereditary property
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): weakly hereditary property (of spaces)
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): weakly hereditary property (of spaces)