On the double density spectra of compact spaces

TL;DR

This paper investigates the double density spectrum of compact spaces, establishing that for polyadic spaces and locally compact groups, the spectrum forms a continuous interval between density and weight.

Contribution

It proves that the double density spectrum equals the interval from density to weight for polyadic spaces and locally compact groups, extending understanding of their topological structure.

Findings

dd(X) = [d(X), w(X)] for polyadic spaces

dd(G) = [d(G), w(G)] for locally compact groups

Provides conditions under which certain cardinals belong to dd(X)

Abstract

The set $dd(X)$ of densities of all dense subspaces of a topological space $X$ is called the double density spectrum of $X$. In this note we present a couple of results that imply $\lambda \in dd(X)$, provided that $X$ is a compact space and $\lambda$ is a cardinal satisfying certain conditions. As a consequence of these results, we prove that $dd(X) = [d(X), w(X)]$ holds for any polyadic space $X$. This, in turn, implies that $dd(G) = [d(G), w(G)]$ for any locally compact topological group $G$.