Analytic Baire spaces. (English) Zbl 1273.54027

Ostaszewski, A. J.

Fundam. Math. 217, No. 3, 189-210 (2012).

The paper builds on concepts and proofs already available in the literature for the Baire space property in the category of classically analytic spaces. Levi’s well known Open Mapping Theorem and Levi’s Comparison Theorem have been proved for separable spaces. The author of this article gives non-separable generalizations of these results.
More precisely, he considers analytic spaces and obtains the main Theorem 1.6 (non separable Levi Open Mapping Theorem) and Corollary 1.7 (Generalized Levi Comparison Theorem).
Also, a non-separable version of a classical result for abelian locally compact groups due to Ellis is presented (Main Theorem 1.9). In this version it is not assumed that the group is abelian or that it is locally compact. Since in the non-separable context continuity is not enough to preserve analyticity an additional condition for the index-\(\sigma\)-discreteness is needed.
The paper is well written. The remarks and the proofs are sufficiently detailed to allow the reader to follow without difficulty. This is a very interesting contribution to Baire space theory. The reader is referred to the paper for details and bibliography.

Reviewer: Nikolaos Papanastassiou (Athens)

MSC:

54E52	Baire category, Baire spaces
26A03	Foundations: limits and generalizations, elementary topology of the line
22A05	Structure of general topological groups
54H05	Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)

Keywords:

automatic continuity; analytic sets; analytic Baire theorem; analytic Cantor theorem; Nikodym’s stability theorem; non-separable descriptive topology; shift-compactness; non-commutative groups; group-norm; open mapping