This is a continuation of an MSE question which received a partial answer (see below).

Given a topological space $\mathcal{X}$, let $C(\mathcal{X})$ be the ring of real-valued continuous functions on $\mathcal{X}$ (with the ring operations given pointwise as usual). For $\varphi$ a first-order sentence in the language of rings, write "$\mathcal{X}\models_C\varphi$" if $C(\mathcal{X})\models \varphi$ in the usual sense, and say that $\varphi$ is **$C$-satisfiable** iff this holds for some $\mathcal{X}$.

My question is:

What is the least cardinal $\kappa$ such that for every $C$-satisfiable sentence $\varphi$ there is some space $\mathcal{X}$ with $\mathcal{X}\models_C\varphi$ and $\vert\mathcal{X}\vert<\kappa$?

Note that we can characterize disconnectedness by a single sentence; for example, either "$\exists x,y(x+y=1\wedge xy=0)$" or "$\exists x(x^2=x\wedge x\not=0\wedge x\not=1)$" will do the trick. This has two consequences. First, it means that the "subspace" analogue of $\kappa$ does not exist at all since there are connected spaces of arbitrarily large cardinality with no connected subspaces of strictly smaller cardinality but still having more than one point (take the Dedekind completion of an appropriately homogeneous dense linear order of large cardinality). Second, with a bit more thought it leads to a proof that $\kappa>2^{\aleph_0}$ (pointed out by Atticus Stonestrom at MSE).

However, I don't see how to go any further than that at the moment.

Ultimately I'm interested in what happens when we replace the ring of real numbers with the usual topology, $\mathbb{R}$, with an **arbitrary topological structure** (= all functions continuous, all relations closed): given a first-order signature $\Sigma$ and a topological $\Sigma$-structure $\mathfrak{A}=(A,\tau,\Sigma^\mathfrak{A})$ we get a natural way to assign to each topological space $\mathcal{X}$ a $\Sigma$-structure with underlying set $C(\mathcal{X}, (A,\tau))$ and so a corresponding satisfaction relation $\mathcal{X}\models_{C:\mathfrak{A}}\varphi$. I'd like to understand how the choice of "target" $\mathfrak{A}$ impacts the resulting abstract model theoretic properties of $\models_{C:\mathfrak{A}}$. But already the case $\mathfrak{A}=\mathbb{R}$ seems nontrivial.

bijectiveon, I believe, locally compact Hausdorff spaces, furnishing Gelfand duality with commutative $C^*$-algebras. So if you are willing to incorporate this structure, you can sort of eliminate the topology altogether. $\endgroup$compactHausdorffs, and the nonunital ones are what you wrote. I now realize that I don't actually know what kind of duality involves all continuous functions on a locally compact Hausdorff space. I wonder if to capture these one should replace one-point-compactification with the Stone-Čech and/or extend the class of spaces to all completely regulars? $\endgroup$8more comments