Elementary chains and compact spaces with a small diagonal

TL;DR

This paper investigates properties of compact spaces with a small diagonal, using elementary chains, and explores conditions under which such spaces are metrizable or have points of countable character, addressing an open problem in topology.

Contribution

It introduces new techniques using elementary chains to analyze compact spaces with a small diagonal and generalizes known results about their structure.

Findings

ccc subspaces have countable -weight

generalization of Gruenhage's result on metrizable fiberings

if a Luzin set exists, such spaces have many points of countable character

Abstract

It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such spaces have countable \pi-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.