L-spaces exist in ZFC; S-spaces consistently do not

A regular space is an S-space if it is hereditarily separable but not Lindelöf, and an L-space if it is hereditarily Lindelöf but not separable. Since separability and the Lindelöf property are dual in most elementary respects, the historical expectation was that the two existence questions would have the same answer. They do not.

(a) An L-space exists in ZFC. Justin Moore (announced 2005, published 2006) constructs one outright, with no extra axiom. The space is a topology $\tau[X]$ on an uncountable set $X \subseteq \omega_1$ of countable ordinals, built from a colouring obtained by analysing oscillations of a coherent sequence $\langle e_\alpha : \alpha < \omega_1 \rangle$ of finite-to-one functions, along the lower trace of Todorcevic's minimal walks on countable ordinals.

(b) It is consistent that no S-space exists. Todorcevic proved that the proper forcing axiom implies there are no S-spaces. Since PFA is consistent relative to a supercompact cardinal, "there are no S-spaces" is consistent relative to that large-cardinal hypothesis.

(c) S-spaces do exist under other hypotheses, for instance under CH, so (b) is a genuine independence and not a theorem.

• Not proved in this library. Neither construction is carried out here. The library now develops separability, the Lindelöf property and their hereditary forms, but it does not develop the minimal-walk and oscillation machinery for the ZFC L-space or the forcing machinery for the S-space consistency result.

• What would prove it. For (a), the combinatorics of minimal walks on countable ordinals and oscillation theory, which is ordinary ZFC but rests on the ordinal machinery and on a substantial theory of colourings. For (b), the proper forcing axiom, hence proper forcing, iteration with countable support, and a supercompact cardinal for its consistency.

• Why it matters here. It is the standard warning against arguing by duality in general topology. "Hereditarily separable" and "hereditarily Lindelöf" look like mirror images, and the corresponding existence questions are not mirror images at all: one is settled outright in ZFC and the other is independent. Any page in this library that states such a duality must therefore state it for the specific properties proved, and never as a general principle. Note that the countability notions the library does have (Finite, countably infinite, countable, uncountable ↗, $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874) ↗) are untouched by this: nothing here is a statement about $\mathbb{R}$, whose separability and Lindelöf property are both elementary.

• Conditional discipline. (a) is a ZFC theorem, cited and not proved. (b) is a relative consistency statement whose hypothesis is strictly stronger than Con(ZFC). Nothing here asserts PFA or the existence of a supercompact cardinal.