Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Preservation of some topological properties under forcing

Chris Lambie-Hanson, Pedro Marun

TL;DR

This work investigates the preservation of topological game-theoretic properties under forcing extensions. By leveraging strong properness techniques, it establishes that II’s winning strategies for the strong countable fan tightness and Rothberger games are preserved in all forcing extensions, significantly strengthening previous indestructibility results. However, the same preservation does not extend to the countable fan tightness or Menger games, and concrete counterexamples illustrate the limits of the approach. The paper also discusses the limitations of strong properness in low-cardinal settings and poses open questions regarding models with differing ground-model and forcing-extension behaviors, notably under $0^{\#}$ or PFA. The results connect forcing theory with fine-grained topological games, offering a robust framework for understanding indestructibility of these properties.

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.

Preservation of some topological properties under forcing

TL;DR

This work investigates the preservation of topological game-theoretic properties under forcing extensions. By leveraging strong properness techniques, it establishes that II’s winning strategies for the strong countable fan tightness and Rothberger games are preserved in all forcing extensions, significantly strengthening previous indestructibility results. However, the same preservation does not extend to the countable fan tightness or Menger games, and concrete counterexamples illustrate the limits of the approach. The paper also discusses the limitations of strong properness in low-cardinal settings and poses open questions regarding models with differing ground-model and forcing-extension behaviors, notably under or PFA. The results connect forcing theory with fine-grained topological games, offering a robust framework for understanding indestructibility of these properties.

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.
Paper Structure (5 sections, 6 theorems, 10 equations)

This paper contains 5 sections, 6 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strong Properness
  4. Games
  5. Open questions

Key Result

Lemma 2.4

Let $(X,\tau)$ be a topological space. Suppose that player II has a winning strategy in the Rothberger (respectively Menger, strong countable fan tightness, countable fan tightness) game. Then $(X,\tau)$ is Rothberger (respectively is Menger, has strong countable fan tightness, has countable fan tig

Theorems & Definitions (23)

  • Example 1.1: giltonPreservationTopologicalProperties2025
  • Example 1.2: Burke, tallCardinalityLindelofSpaces1995
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • Lemma 3.2: Gilton-Neeman
  • proof
  • ...and 13 more