Hereditary, open-hereditary and closed-hereditary properties of topological spaces

A property of topological spaces is a condition $P$ that is either true or false of each space, as in Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological; a topological property is one whose truth value is the same for homeomorphic spaces. Every subset of a space is regarded as a space by giving it the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

Let $P$ be a property of topological spaces. Then $P$ is

• hereditary if, whenever a space $X$ has $P$, every subspace of $X$ has $P$;
• open-hereditary if, whenever $X$ has $P$, every subspace $S \subseteq X$ with $S$ open in $X$ has $P$;
• closed-hereditary if, whenever $X$ has $P$, every subspace $S \subseteq X$ with $S$ closed in $X$ has $P$.

A hereditary property is both open-hereditary and closed-hereditary, since the condition on $S$ is only a restriction of the range of subspaces quantified over. Neither of the two weaker notions implies the other, and neither implies heredity.

The definition is stable under the route by which a subspace is reached. If $S \subseteq T \subseteq X$ then the topology $S$ inherits from the subspace $T$ is the topology $S$ inherits from $X$, transitivity being discharged in Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace. So "every subspace of $X$" is unambiguous, and a hereditary property automatically passes from $X$ to a subspace of a subspace, with no separate induction.

Heredity is a statement about a property, not about a space. It quantifies over all spaces having $P$ and all their subspaces, so a single space whose subspaces all inherit $P$ says nothing; and a single space that has $P$ and has one subspace lacking $P$ refutes heredity outright. A space that lacks $P$ refutes nothing, however its subspaces behave. That asymmetry is why the failures are recorded here as counterexamples and the successes as theorems.

Only topological properties are worth asking about. Taking $S = X$ shows that a hereditary property holds of $X$ itself, and the subspace topology on $X$ is $\mathcal{T}$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, with $U \cap X = U$), so the definition is not vacuous at the top. But a condition that is not invariant under homeomorphism can be hereditary for uninteresting reasons, since a subspace is only determined up to the identification of its topology (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison); every property named hereditary in this library is a topological property, and it is said so where it is proved.

• The three notions separate in practice. Metrizability and first countability are hereditary, and that is proved in the next item. "Has a countable dense subset" is open-hereditary, by claim 4 of For $A \subseteq S \subseteq X$ the closure of $A$ in $S$ is $\overline{A}^{X} \cap S$, while the interior only contains $\operatorname{int}^{X}(A) \cap S$, with equality when $S$ is open; and a dense subset of $X$ traces to a dense subset of every open $S$, and is not hereditary; the witness is worked on the companion page, where an uncountable discrete subspace is exhibited inside a space that has a countable dense subset.

• What is deliberately not settled here. Whether the separation properties beyond the Hausdorff condition of Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not are hereditary is a question about axioms that are not available at this point in the reading order, and no claim about them is made on this page.

• Products have their own word. A property preserved by arbitrary products is usually called productive, and the same three-way refinement (finite products, countable products, arbitrary products) applies to it. No item on this page uses that word, because the productive theorems it would organise are not available at this point in the reading order.