L-spaces, S-spaces, and strong S-spaces

A topological space is hereditarily separable when every one of its subspaces is separable, and it is hereditarily Lindelöf when every one of its subspaces is Lindelöf. Here every subset carries its subspace topology, as in Hereditary, open-hereditary and closed-hereditary properties of topological spaces; separability and Lindelöfness have the meanings in Separability: the existence of an at most countable dense subset and Countably compact, Lindel"of, sequentially compact, limit point compact and $\sigma$-compact spaces, and relatively compact subsets.

Using this library's convention that regularity does not itself include any separation axiom, a space is

• an L-space if it is regular and Hausdorff, hereditarily Lindelöf, and nonseparable;
• an S-space if it is regular and Hausdorff, hereditarily separable, and not Lindelöf; and
• a strong S-space if every nonempty finite power is an S-space.

Thus a strong S-space is itself an S-space (take the first power), and every finite power of a strong S-space is hereditarily separable. Empty powers are not part of the definition: the zeroth power is a singleton, hence Lindelöf and not an S-space, so including it would make the notion impossible.

The regular-plus-Hausdorff clause implies $T_3$ under the conventions of Regular spaces and $T_3$ spaces, with the source disagreement over whether regularity includes $T_1$ stated explicitly and Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not. It is nevertheless written in the literature's customary form so that regularity is not silently given a different meaning.

Some sources call any regular Hausdorff, hereditarily separable but not hereditarily Lindelöf space an S-space. The two conventions have the same existence content: under that broader wording, choose a subspace that is not Lindelöf; regularity, Hausdorffness, and hereditary separability pass to that subspace, which is an S-space in the definition above. Conversely, a space that is not Lindelöf is certainly not hereditarily Lindelöf. We keep the narrower definition rather than silently exchanging the two statements.

These definitions make no choice. Later assertions that construct examples simultaneously along ${\omega }_1$, or that use PFA or CH in ZFC, declare those axioms at the point of use.