All Parovichenko spaces may be soft-Parovichenko

Abstract

It is shown that, assuming the Continuum Hypothesis, compact Hausdorff space of weight at most $\mathfrak{c}$ is a remainder in a soft compactification of $\mathbb{N}$. We also exhibit an example of a compact space of weight ${\aleph }_1$ -- hence a remainder in some compactification of $\mathbb{N}$ -- for which it is consistent that is not the remainder in a soft compactification of $\mathbb{N}$.

Version 2: simpler construction, plus a remainder of $\mathbb{N}$ that is consistently not a soft remainder of $\mathbb{N}$ Version 3: new author; examples of soft-Parovichenko spaces as well as some questions Version 4: new title, corrections after referee's report