Let M=N×[0,1]. The natural projection π:M→N, which sends (n,x) to n, induces a projection mapping π⁎:M⁎→N⁎, where M⁎ and N⁎ denote the Čech-Stone remainders of M and N, respectively. We show that CH implies every autohomeomorphism of N⁎ lifts through the natural projection to an autohomeomorphism of M⁎. That is, for every homeomorphism h:N⁎→N⁎ there is a homeomorphism H:M⁎→M⁎ such that π⁎∘H=h∘π⁎. This complements a recent result of the second author, who showed that this lifting property is not a consequence of ZFC. Combining this lifting theorem with a recent result of the first author, we also prove that CH implies there is an order-reversing autohomeomorphism of H⁎, the Čech-Stone remainder of the half line H=[0,∞).

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.

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

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.

The Proper Forcing Axiom implies that compact Hausdorff spaces are either first-countable or contain a converging ω1-sequence.

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).

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.

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.