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The $\rho_d$ are indeed seminorms; this can be verified by direct calculations or by noting that $\sup\lbrace |g(x,y)|:|x-y|=d, x,y\in[a,b]\rbrace$ defines a seminorm on $C([a,b]^2,\mathbb{R})$ (as …

Another way of constructing the space is as follows: for every finite subset $F$ of $X$ let $Y_F=\beta (X\setminus F)$. If $F\subseteq G$ then there is a natural map $f^G_F:Y_G\to Y_F$; this gives us …

It implies the Banach-Tarski Paradox ...

Denote the intersection of a closed linear subspace $A$ with the unit sphere by $S_A$, say. You can define the distance of $A$ and $B$ to be the Hausdorff distance between $S_A$ and $S_B$. This will g …

Kuratowski proved in Sur les théorèmes du „plongement" dans la théorie de la dimension. Fundamenta Mathematicae 28.1 (1937): 336-342 that the set of embeddings of an at most $n$-dimensional separable …

I'm using the notation of the original paper. In hindsight one could have used the principal ultrafilter generated by $\{n\}$, where $n$ is the index of the extension $F$ found in the proof. But that …

This paper, Rigid Stone spaces within $\mathsf{ZFC}$ by Dow, Gubbi, and Szymański contains relatively easy examples of such spaces.

If the interior of $X$ were nonempty there would be nonempty open sets $U$ and $V$ in $L^1$ such that $U\times V\subseteq X$. But then $U\subseteq A$ shows that $A$ would have nonempty interior. Note …

This characterizes normality. That it implies normality was observed above by Remy. Conversely, assume $f$ vanishes on $A\cap B$. Define $h:A\cup B\to\mathbb{R}$ by $h(x)=f(x)$ if $x\in A$ and $h(x)=0 …

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 $X$.

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 …