Shift-compactness in almost analytic submetrizable Baire groups and spaces (Q2850624)

This is a survey based on a long list of recent papers by the author, many in collaboration with N. H. Bingham and one with H. I. Miller. It concerns mainly the concepts of normed group, shift-compactness, almost completeness and analyticity, with the aim to underline the interplay among them, in the category of (sub)metrizable spaces. We briefly describe the contents of the paper.NEWLINENEWLINEThe definition of normed group is recalled, and also its historical background and an inspiring ``principal example of a normed group''. A theorem gives equivalent conditions for a normed group to be a topological group, and consequences of this result are listed. Several properties of normed groups are given; among them, the author quotes the so-called Squared Pettis Theorem, which is applied for the proof of the Steinhaus Subgroup Theorem: let \(G\) be an almost-complete normed group and \(H\) a non-meagre subgroup of \(G\) with the Baire property; if either \(H\) is dense in \(G\) or \(G\) is connected, then \(H=G\). As an application of the latter theorem, a result by Loy and Hoffmann-J{o}rgensen is derived, stating that a non-meagre analytic topological group is Polish.NEWLINENEWLINEThe concept of shift-compactness is discussed, explaining why this is a notion of compactness. Among other known results, the importance of the Shift-compactness Theorem is stressed; it was inspired by the Kastelman-Borwein-Ditor Theorem from classical real analysis. Moreover, shift-compactness is considered in relation with separation properties. Applications of shift compactness are given, in particular, a Baire version of the Effros Open Mapping Principle on the action of a normed group on a separable metrizable space.NEWLINENEWLINEFinally, many results concerning analyticity are recalled, for example analytic versions of the Baire Theorem and the Cantor Theorem, and a characterization of almost completeness for normed groups. Moreover, a condition is given for a normed group to be a Polish topological group (Semipolish Theorem), and also applications to weak category convergence are mentioned.