We present examples of realcompact spaces with closed subsets that are C*-embedded but not C-embedded, including one where the closed set is a copy of N. @en

It is shown that, assuming the Continuum Hypothesis, every compact Hausdorff space of weight at most c is a remainder in a soft compactification of N.

We also exhibit an example of a compact space of weight aleph_1 ---hence a remainder in some compactification of N ---for which it is consistent that is not the remainder in a softcompactification of N.
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The Proper Forcing Axiom implies that compact Hausdorff spaces are either first-countable or contain a converging ω1-sequence.@en

We present an example of a zero-dimensional F-space that is not strongly zero-dimensional.

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We show that the existence of a homeomorphism between ω0⁎ and ω1⁎ entails the existence of a non-trivial autohomeomorphism of ω0⁎. This answers Problem 441 in [8]. We also discuss the joint consistency of various consequences of ω0⁎ and ω1⁎ being homeomorphic.
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We prove that compact Hausdorff spaces with a PP-diagonal are metrizable. This answers problem 4.1 (and the equivalent problem 4.12) from Cascales et al. (2011).
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It is a well known open problem if, in View the MathML sourceZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of substructures. We prove that ccc subspaces of such spaces have countable ππ-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.@en