A compactification, \(\gamma \mathbb{N}\), of the discrete space of all natural numbers is said to be soft if for all pairs \(\langle A, B \rangle\) of disjoint subsets of \(\mathbb{N}\) the following holds: whenever \(\overline{A} \cap \overline{B} \neq \emptyset\) there is an autohomeomorphism \(h\) of \(\gamma \mathbb{N}\) such that \(h[A] \cap B\) is infinite and \(h\) is the identity on the remainder \(\gamma \mathbb{N} \setminus \mathbb{N}\).
Banakh asked (at MathOverFlow) whether every Parovichenko space is soft-Parovichenko, where a Parovichenko space is defined to be a remainder in some compactification of \(\mathbb{N}\) (thus, a soft-Parovichenko space is a remainder in some soft compactification of \(\mathbb{N}\)). Parovichenko’s classic theorem (from [I. I. Parovichenko, Sov. Math., Dokl. 4, 592–595 (1963; Zbl 0171.21301); translation from Dokl. Akad. Nauk SSSR 150, 36–39 (1963)]) characterizes, assuming the Continuum Hypothesis, the Parovichenko spaces to be, precisely, the compact Hausdorff spaces of weight at most \(\mathfrak{c}\).
In the paper under review, it is shown that, assuming the Continuum Hypothesis, every compact Hausdorff space of weight at most \(\mathfrak{c}\) is a remainder in a soft compactification of \(\mathbb{N}\) – that is, the authors have established that, under \(\mathbf{CH}\), Banakh’s question is answered positively.