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Definition: Hereditary and Weakly Hereditary Properties
If every subspace of a topological space $(X,\mathcal O)$ has a property of $X$, this property is said to be hereditary.
A property of $X$ is said to be weakly hereditary, if every closed subset $U\subseteq X,$ considered as a subspace of $X$ has this property.
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Propositions: 1 2
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References
Bibliography
 Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
 Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition