Descriptive compact spaces and renorming

Abstract

Abstract. We study the class of descriptive compact spaces, the Banach spaces generated by descriptive compact subsets and their relation to renorming problems.

On Gruenhage spaces, separating $$\sigma $$ σ -isolated families, and their relatives

“…The following result is already known [20,23,25] and shows the behaviour of descriptive compacta, Gruenhage spaces and fragmentable ones. Proposition 1.2.…”

“…Descriptive compacta were introduced by Hansell [6] and comprehensively studied in [20], and connections with LUR renorming theory can be found in [23]. The first proof of the fact that any descriptive compact is Gruenhage compact is based on the characterization of descriptive compacta in terms of property SLD due to Oncina and Raja in [20]. On the other hand, the notion of fragmentable space was introduced by Jayne and Rogers in [11] in connection with Borel selectors of multivalued functions.…”

“…These are the class of descriptive compacta and that of fragmentable spaces. Descriptive compacta were introduced by Hansell [6] and comprehensively studied in [20], and connections with LUR renorming theory can be found in [23]. The first proof of the fact that any descriptive compact is Gruenhage compact is based on the characterization of descriptive compacta in terms of property SLD due to Oncina and Raja in [20].…”

“…Let us consider a σ-isolated network N on X. By Lemma 2.5 and the regularity hypothesis, it is not restrictive to assume that N ⊂ F ∩ G; see [20] for more details. Then, for every n < ω, there exist F n ∈ F and G n ∈ G such that ∪N n = F n ∩ G n .…”

Descriptive classes of sets in nonseparable spaces

“…There are further results on the coincidence of Borel sets in the norm and weak topologies, see [55,Proposition 1.5] or [56,Corollary 2.7].…”