Hereditary Property - in Topology

In Topology

In topology, a topological property is said to be hereditary if whenever a topological space has that property, then so does every subspace of it. If the latter is true only for closed subspaces, then the property is called weakly hereditary.

For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff compactness are weakly hereditary, but not hereditary. Connectivity is not weakly hereditary.

If P is a property of a topological space X and every subspace also has property P, then X is said to be "hereditarily P".