*This was previously asked and bountied at math.stackexchange with no response.*

Let $C(\mathbb{R}^2,\mathbb{R})$ be the space of all continuous functions $\mathbb{R}^2\rightarrow \mathbb{R}$ with the compact-open topology. 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 $\{C\}_\mathbb{R}\cong\{A\}_\mathbb{R}$?

$\{C\}_\mathbb{R}$ is pretty tame since as Eric Wofsey observed it's a vector subspace of $C(\mathbb{R}^2,\mathbb{R})$. However, $\{A\}_\mathbb{R}$ seems much weirder. For example, I don't even know whether $\{A\}_\mathbb{R}$ is connected.

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