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 :
 * $\map \xi X \implies \map \xi Y$

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)