Indestructibility of compact spaces

TL;DR

This paper explores the stability of compact spaces under certain forcing conditions and examines the relationships between topological properties and set-theoretic assumptions, including the Continuum Hypothesis.

Contribution

It demonstrates that under CH, certain generalizations of the Rothberger property are not equivalent for compact spaces and links the $eth_1$-Borel Conjecture to large cardinal assumptions.

Findings

Generalizations of the Rothberger property are not equivalent under CH.

The $eth_1$-Borel Conjecture is equiconsistent with an inaccessible cardinal.

Compactness can be preserved or not under countably closed forcing depending on set-theoretic assumptions.

Abstract

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to $\omega_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba's "$\aleph_1$-Borel Conjecture" is equiconsistent with the existence of an inaccessible cardinal.