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