Hereditarily operator

Definition

Let $p$ be a property of topological spaces. The property hereditarily $p$ is defined as the property of being a topological space in which every subspace has property $p$ (when given the subspace topology).

Relation with other property modifiers

• Weakly hereditarily operator is the property modifier that imposes the condition only on closed subspaces (viz, closed subsets)

Related topological space metaproperties

A topological space property is termed subspace-closed, or subspace-hereditary, if it is invariant under the hereditarily operator, viz any subspace of a space with the property, also has the property.