Nonseparable growth of the integers supporting a measure

TL;DR

This paper constructs a specific compactification of the natural numbers where the remainder set is both nonseparable and supports a positive measure, under certain set-theoretic assumptions.

Contribution

It introduces a method to create a compactification with a nonseparable measure-supporting remainder, assuming $rak b = rak c$ or weaker conditions.

Findings

The remainder set is nonseparable.

The remainder supports a strictly positive measure.

Construction relies on set-theoretic assumptions.

Abstract

Assuming $\mathfrak b = \mathfrak c$ (or some weaker statement), we construct a compactification $\gamma\omega$ of $\omega$ such that its remainder $\gamma\omega\setminus\omega$ is nonseparable and carries a strictly positive measure.