Compact spaces, lattices, and absoluteness: a survey

TL;DR

This survey examines how certain classes of compact spaces, like Eberlein and Gul'ko compacta, remain stable across set-theoretic universe extensions, while others like Corson compacta may not.

Contribution

It provides a comprehensive overview of the absoluteness and stability of various compact space classes under set-theoretic extensions.

Findings

Eberlein/Gul'ko compacta are stable across extensions

Corson compacta may lose their properties in forcing extensions

The survey consolidates results from the author's notes (2004-2006)

Abstract

Given a compact space in a fixed universe of set theory, one can naturally define its interpretation in any ZFC extension of the universe. We investigate the stability of some classes of compact spaces with respect to extensions of this sort. We show that the class of Eberlein/Gul'ko compacta is stable (= absolute). On the other hand, there are examples of Corson compacta which are no longer Corson in some forcing extensions. All the material comes from the author's notes written between 2004 -- 2006.