Preservation of some topological properties under forcing

TL;DR

This paper investigates how certain topological properties, specifically strong countable fan tightness and the Rothberger property, are preserved under set forcing extensions, providing new insights into their robustness in set-theoretic topology.

Contribution

It proves that strong countable fan tightness and the Rothberger property are preserved under all set forcing extensions, answering a question posed by Gilton and Holshouser.

Findings

Strong countable fan tightness is preserved under set forcing.

The Rothberger property is preserved under set forcing.

Countable fan tightness and Menger games are not necessarily preserved.

Abstract

We add to the theory of preservation of topological properties under forcing. In particular, we answer a question of Gilton and Holshouser in a strong sense, showing that if player II has a winning strategy in the strong countable fan tightness game of a space at a point, then this continues to hold in every set forcing extension of the universe. The same is also true for the Rothberger game, but not for the countable fan tightness or Menger games.