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In this paper, Dimension phenomena associated with $\beta\mathbb{N}$-spaces, Ilijas Farah proved that continuous maps from $\beta\mathbb{N}^2$ (and other powers) to $\beta\mathbb{N}$ are quite simple: …

0a Correct, except for one point: is $X$ is not completely regular then there is no $\beta X$, so the third equivalence in `functionally Hausdorff' does not exist. 0b Not quite, the paper mentioned i …

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 …

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 …

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 …

The spaces in this answer are pseudocompact.

No. There is a compactification of $\mathbb{N}$ whose remainder, $K$, is ccc non-separable. So there is a continuous surjection $f$ from $\beta\mathbb{N}\setminus\mathbb{N}$ onto $K$; take a closed su …

Let $X_n$ be $\{k\in\mathbb{N}:k\ge n\}$ and let $f_n:X_{n+1}\to X_n$ be the inclusion map. The inverse limit of the system $\{X_n,f_n,\mathbb{N}\}$ is empty; the limit of the system $\{\beta X_n,\bet …

$\beta\mathbb{N}$ 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 …

Topologically speaking $\mathbb{Z}$ with the topology mentioned above is just (homeomorphic to) the space of rational numbers. The space $\beta\mathbb{Q}$ has been studied a lot (not as much as $\beta …

More generally: if $X$ is normal and $A$ is closed in $X$ then, by the Tietze-Urysohn theorem, the closure in $\beta X$ of $A$ is $\beta A$. In the example above $X=\mathbb{R}$ and $A=\mathbb{R} \setm …

Not without further assumptions. First create an ordered space $X$ by identifying $\langle0,\omega_1\rangle$ and $\langle1,\omega_1\rangle$ in the product $2\times(\omega_1+1)$ to a point, $\Omega$ sa …

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 …

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 …