*This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing answers were aimed at the earlier version of the question.*

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, and consider the following two subspaces:

$\mathcal{Asso}$ = the subspace of all

**associative**functions.$\mathcal{Comm}$ = the subspace of all

**commutative**functions.

Of course these are distinct as sub*sets*; my question is whether they are topologically distinguishable:

Is $\mathcal{Asso}\cong\mathcal{Comm}$?

I strongly suspect that the answer is negative - in particular, $\mathcal{Comm}$ is connected and I suspect $\mathcal{Asso}$ is not - but I don't see how to prove this. (Another nice feature of $\mathcal{Comm}$ is that it is an $\mathbb{R}$-vector space in a natural way; however, it's not clear how to extract any power from this observation.)

Generally, $\mathcal{Asso}$ seems rather mysterious, and I'm interested in any information about it.