A topological version of the Lowenheim-Skolem number 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.
 A: In response to Noah's request after my earlier comments:  this doesn't answer the question, but may shed some light.  If $\kappa$ is supercompact, then for every $C$-satisfiable sentence $\phi$, there is a topological space $\mathcal{X}$ of size $<\kappa$ such that $\mathcal{X} \models_C \phi$.  In fact, the entire powerset of (the underlying set of) $\mathcal{X}$---in particular, its topology---can be taken to be of size $<\kappa$.
Proof sketch:  suppose $\mathcal{Y}=(Y,\tau) \models_C \phi$.  Fix a regular $\theta$ such that $\mathcal{Y} \in H_\theta$, where $H_\theta$ denotes the collection of sets of hereditary cardinality $<\theta$ (you could work with rank initial segments if you prefer).
Since $\kappa$ is supercompact, there is some $\mathfrak{N} \prec (H_\theta,\in)$ of size $<\kappa$ such that $\mathcal{Y} \in \mathfrak{N}$ and the transitive collapse $\mathfrak{H}(\mathfrak{N})$ of $\mathfrak{N}$ is equal to $H_\mu$, where $\mu$ is the ordertype of $\mathfrak{N} \cap \theta$.
(This is folklore; see Fact 7.7 of my paper for a proof.  In fact existence of such models capturing all possible sets is equivalent to supercompactness, by a classic theorem of Magidor).
In particular, this ensures that $\mathfrak{H}(\mathfrak{N})$ is correct about lots of things and closed under lots of operations.
For each $b \in \mathfrak{N}$, let $b_{\mathfrak{N}}$ denote the image of $b$ under the transitive collapsing map. By elementarity of the collapsing map and the fact that the statement ``$\mathcal{Y} \models_C \phi$" is downward absolute from the universe to the structure $(H_\theta,\in)$, it follows that

$\mathfrak{H}(\mathfrak{N}) \models$ ``$\Big( Y_{\mathfrak{N}}, \tau_{\mathfrak{N}} \Big)$ is a topological space, and the ring of continuous real-valued functions on it satisfies the formula $\phi_{\mathfrak{N}}$".

Since $\phi$ is just a formula in a countable language, the transitive collapsing map fixes $\phi$, i.e., $\phi = \phi_{\mathfrak{N}}$.   So

$\mathfrak{H}(\mathfrak{N}) \models$ ``$\Big( Y_{\mathfrak{N}}, \tau_{\mathfrak{N}} \Big)$ is a topological space, and the ring of continuous real-valued functions on it satisfies the formula $\phi$".

Now $\mathfrak{H}(\mathfrak{N})$, being of the form $H_\mu$, is correct about the quoted statement (this is standard stuff; in fact, in this particular case the quoted statement is even $\Pi_1$, and sets of the form $H_\mu$ are always $\Pi_1$ elementary in the universe).
And the space $\Big( Y_{\mathfrak{N}}, \tau_{\mathfrak{N}} \Big)$ is of size $\le |\mathfrak{H}(\mathfrak{N})|= |\mathfrak{N}|<\kappa$.
