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