All Parovichenko spaces may be soft-Parovichenko
Keywords:
compactification, soft compactification, Parovichenko spaces, Continuum HypothesisAbstract
To the memory of Phil Zenor, one of the founders of this journal
Abstract. 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 $\mathbb{N}$.
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 $\mathbb{N}$.
References
Taras Banakh, Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $mathbb{N}$ ?, https://mathoverflow.net/q/309583. (version: 2019-08-18).
Taras Banakh and Igor Protasov, Constructing a coarse space with a given Higson or binary corona, Topology and its Applications 284 (2020), 107366, 20, DOI 10.1016/j.topol.2020.107366. MR4142223
Aleksander Błaszczyk and Andrzej Szymański, Concerning Parovičenko's theorem, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques 28 (1980), no. 7-8, 311-314 (1981) (English, with Russian summary). MR628044
Alan Dow, On compact separable radial spaces, Canadian Mathematical Bulletin 40 (1997), no. 4, 422-432, DOI 10.4153/CMB-1997-050-0. MR1611327
R. Engelking and M. Karłowicz, Some theorems of set theory and their topological consequences, Fundamenta Mathematicae 57 (1965), 275-285, DOI 10.4064/fm-57-3-275-285. MR196693
S. P. Franklin and M. Rajagopalan, Some examples in topology, Transactions of the American Mathematical Society 155 (1971), 305-314, DOI 10.2307/1995685. MR283742
I. I. Parovičenko, A universal bicompact of weight $aleph$, Soviet Mathematics Doklady 4 (1963), 592-595. Russian original: Ob odnom universal' nom bikompakte vesa $aleph$, Doklady AkademiY Nauk SSSR 150 (1963) 36-39. MR0150732 (27#719)
