A definitive improvement of a game-theoretic bound and the long tightness game

TL;DR

This paper proves a new cardinal inequality for spaces with points G_delta using an extended Menger game, improving previous results and discussing a long version of the tightness game.

Contribution

It provides a full proof of a cardinal inequality for G_delta point spaces using a long Menger game, extending prior work under the Continuum Hypothesis.

Findings

Established a cardinal inequality for G_delta point spaces.

Extended the Menger game to a long version for new insights.

Discussed a long version of the tightness game.

Abstract

The main goal of the paper is the full proof of a cardinal inequality for a space with points $G_\delta $, obtained with the help of a long version of the Menger game. This result, which improves a similar one of Scheepers and Tall, was already established by the authors under the Continuum Hypothesis. The paper is completed by few remarks on a long version of the tightness game.