*This was previously [asked and bountied](https://math.stackexchange.com/q/3619809/28111) 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.*

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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.