All Parovichenko spaces may be soft-Parovichenko

TL;DR

Under the Continuum Hypothesis, all Parovichenko spaces can be embedded as remainders in soft compactifications of the natural numbers, but some spaces of weight  may not admit such a soft compactification.

Contribution

The paper demonstrates that assuming CH, all Parovichenko spaces are soft-Parovichenko, and provides an example of a space that cannot be embedded as a soft remainder, highlighting limitations.

Findings

All Parovichenko spaces are soft-Parovichenko under CH.

Existence of a space of weight  not embeddable as a soft remainder.

Conditional nature of soft compactifications based on space weight.

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 $\mathbb{N}$.