Definition:Weakly Hereditary Property

Definition

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

Then $\xi$ is weakly hereditary iff:

$\xi \left({X}\right) \implies \xi \left({Y}\right)$

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

• Hereditary property

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$