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The statement is analogous to the general result that, for Tychonoff spaces, compactness is equivalent to pseudo-compactness plus realcompactness, see the beginning of section 3.11 in Engelking's Gene …

No, your $\beta_2S$ is $\beta S$, the Čech-Stone compactification of $S$. As an example let $S=\omega$ and $P=\omega+1$ (with order topology, so it is just the converging sequence). Let $g$ be the cha …

A partial answer. for your specific example, the algebra $B$ generated by the rational intervals, the answer is negative. This algebra is countable and dense in $2^\mathbb{R}$. It is countable and ze …

Yes if the point is from $\mathbb{N}$ (it is isolated). No if the point is in $\beta\mathbb{N}\setminus\mathbb{N}$ because in that subspace every nonempty $G_\delta$-set has nonempty interior, see thi …

You need to assume more. Here's an example let $T=\omega$, the set of finite ordinals and let $U=\omega+1$ with its order topology (`the' convergent sequence). Define $f:T\times\{0,1\}\to U$ by $f(t,0 …

For an explicit pair of disjoint open sets with intersecting closures work on the binary tree $2^{<\omega}$ of finite sequences of $0$s and $1$s. For every $x\in2^\omega$ let $A_x=\{x\mathbin{\upharpo …

Since $X$ is discrete every prime ideal in $C_b(X)$ is maximal and so $\operatorname{Spec}C_b(X)$ is just $\beta X$ and you already have two dual extensions of the operation on $X$.

The family $\mathcal{F}=\{ F\in \mathcal{R}( \beta X) :p\in \operatorname{int}_{\beta X}F\}$ is indeed a filterbase; it is a base for the neighbourhood filter at $p$. As noted in the comments there ne …

The first part of the family, $\{F:p\in\mathrm{int}_{\beta X}F\}$ is a family of neighbourhoods of $p$. Each $A_n$ is the closure of its interior (the $R$ suggests we have regular closed sets here). T …

$\beta\mathbb{N}$ is not a retract of a Tychonoff cube because it is not connected; it also not a retract of a Cantor cube, not even a continuous image, see problem 3.12.12 in Engelking's book. It is …

There is, in general, a on-to-one correspondence between closed subalgebras of $C^*(X)$ (the algebra of bounded continuous real-valued functions) and the compactifications of $X$. Closed in the sense …

Comments from Alan Dow: (1) under CH the set of P-points is dense and radial even, using $\omega_1$-sequences. (2) if $D$ is dense and pseudoradial then for every cozero set $C$ of $\omega^*$ the inte …

As an answer to the bonus question: no, see K. P. Hart and J. Vermeer. Fixed-point sets of autohomeomorphisms of compact F-spaces, Proceedings of the American Mathematical Society, 123 (1995), 311–314 …

Answer to 1: In On closed subspaces of $\omega^*$ (Proc. AMS, 1993) it is shown by Dow, Frankiewicz and Zbierski that in the $\aleph_2$-Cohen model every compact zero-dimensional $F$-space of weight a …

A partial answer: other examples of pretty compact spaces are uncountable powers of $\{0,1\}$ and $[0,1]$, and in general products of uncountably many non-trivial compact Hausdorff spaces. See Problem …