Hereditary Property

A property that passes from a topological space to every subspace with respect to the relative topology.

Examples are first and second countability, metrizability, the separation axioms , and , and some of the related properties, such as the one of being a regular, completely regular, or Tychonoff space.

Axiom  is not hereditary, nor is normality, though counterexamples (such as the Tychonoff plank) are hard to find). It is much easier to find disconnected subspaces of connected subspaces (such as, for example, a union of two disjoint disks in the Euclidean plane; left figure) or non-compact subspaces of compact subspaces (e.g., an open disk inside a closed disk; right figure).

REFERENCES:

Kelley, J. L. General Topology. New York: Van Nostrand, p. 133, 1955.