Gruenhage spaces and their influence on Banach space renorming theory

TL;DR

This paper surveys Gruenhage spaces, a class of topological spaces, and explores their significant influence on Banach space renorming theory, highlighting key developments since their introduction.

Contribution

It provides a comprehensive overview of Gruenhage spaces and their impact on the evolution of renorming theory in Banach spaces.

Findings

Gruenhage spaces are influential in Banach space renorming.

They helped solve problems related to dense G_delta subsets.

The survey clarifies the role of these spaces in functional analysis.

Abstract

In a paper from 1987, Gruenhage defined a class of topological spaces that now bear his name, and used it to solve a problem of Talagrand on the existence of dense $G_\delta$ metrizable subsets of Gul'ko compact spaces. Gruenhage's paper became highly influential among researchers in renorming theory, a branch of Banach space theory. In this paper we survey Gruenhage and related spaces, and their interactions with renorming theory.