Densely k-separable compacta are densely separable

TL;DR

This paper investigates the relationship between densely k-separable compact spaces and their separability, revealing that densely k-separable compacta are necessarily densely separable, linking topological properties with cardinal invariants.

Contribution

It establishes that densely k-separable compact spaces are densely separable, connecting this property with known cardinal invariants like $ au$-weight and $ ho$-weight.

Findings

Densely k-separable compacta are densely separable.

The property implies all dense sets are separable in compact spaces.

Connections between $ au$-weight, $ ho$-weight, and density are elucidated.

Abstract

A space has $\sigma$-compact tightness if the closures of $\sigma$-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known connections established between $\pi$-weight and the density of all dense subsets, or more precisely, the cardinal invariant $\delta(X)$.