Open-point topological games and productivity of dense-separable property

TL;DR

This paper explores open-point topological games to understand the density bounds of dense subsets and their behavior under products, with applications to Banach spaces possessing the (CSP) property.

Contribution

It introduces new methods using topological games to analyze density invariants and their product behavior, applying these to Banach spaces with the (CSP) property.

Findings

Established cardinal equalities and inequalities for density bounds.

Analyzed the behavior of density invariants in product spaces.

Applied results to Banach spaces with the (CSP) property.

Abstract

In this note we study the open-point topological games in order to analyze the least upper bound for density of dense subsets of a topological space. This way we may also analyze the behavior of such cardinal invariants in taking products of spaces. Various related cardinal equalities and inequalities are given. As an application we take a look at Banach spaces with the property (CSP) which can be formulated by stating that each weak-star dense linear subspace of the dual is weak-star separable.