Let $C(\mathbb{R}^2,\mathbb{R})$ be the space$^1$ of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$. I'm interested in analyzing subspaces of $C(\mathbb{R}^2,\mathbb{R})$ determined by first-order theories, especially finite equational theories. Specifically, for $T$ a set of first-order sentences in the language of a single binary operation, let $T_\mathbb{R}$ be the subspace of continuous $f$ such that $(\mathbb{R};f)\models T$. I'm curious about how $T$ affects the purely topological properties of $T_\mathbb{R}$. For example, when is $T_\mathbb{R}$ connected? etc.
However, even in very concrete cases, I'm having trouble understanding what $T_\mathbb{R}$ looks like. Letting $C$ and $A$ be the usual statements of commutativity and associativity, I think an answer to the following question would clear things up immensely:
Question: is $\{C\}_\mathbb{R}\cong\{A\}_\mathbb{R}$?
The answer really should be negative - these are very different properties - but I don't see how to get started.
(Actually, I'm interested in general $C(X^2,X)$ and not just $X=\mathbb{R}$, but this special case seems already very interesting on its own.)
$^1$With respect to the compact-open topology. That said, if there's some other natural topology on $C(\mathbb{R}^2,\mathbb{R})$ (like pointwise-convergence) which gives an interesting answer, I'd be happy with that too.