Effros, Baire, Steinhaus and non-separability

Abstract

Abstract. We give a short proof of an improved version of the E¤ros Open Mapping Principle via a shift-compactness theorem (also with a short proof), involving 'sequential analysis'rather than separability, deducing it from the Baire property in a general Baire-space setting (rather than under topological completeness). It is applicable to absolutely-analytic normed groups (which include complete metrizable topological groups), and via a Steinhaus-type Sum-set Theorem (also a consequence of the shift-compactness theorem) includes the classical Open Mapping Theorem (separable or otherwise).

Beyond Erdős-Kunen-Mauldin: Shift-compactness properties and singular sets

“…[Ost2,4]. As one would expect, this variant of KBD does indeed imply Theorem 2 when specialized to topological groups (see the Theorem quoted from [Ost4] at the end of the section).…”

“…So U x is open, for any open U: In particular, if G in Theorem2 is a topological Baire group acting on itself, that action has the Nikodym property, so the following result implies the conclusion of Theorem 2. [Ost4]. For T a Baire non-meagre subset of a metric space X and G a group, Baire under a right-invariant metric, and with separately continuous and transitive Nikodym action on X:…”

Beyond Haar and Cameron-Martin: The Steinhaus support

“…if H is a subgroup of G, as will be the case in Theorem 3.8 below. (The group-theoretic approach to shiftcompactness is that of a group action, here of translation in G -see [MilO]; for applications see [Ost2].)…”

On the scope of the Effros theorem

“…Finally, we remark that generalizations of Theorem 1.4 to the non-separable realm do exist (see [Os1] and [Os2]), although the situation is not as pleasant as in the separable case.…”