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This question is a follow-up to this old one of mine.

Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in the multivariable sense. Since an element of $\mathcal{A}$ is determined by its behavior on the unit disc $D$, we can give $\mathcal{A}$ a natural metric structure: $$d(\star_1,\star_2)=\iint_D \vert (x\star_1 y)-(x\star_2y)\vert dxdy.$$

Question: Is $\mathcal{A}$, equipped with this metric, locally connected?

The question I'm really interested in is the overall topology of the space of all continuous associative operations on $\mathbb{R}$ (basically one aspect of the above-linked question), but that seems harder than anticipated; I'm hoping that by restricting to very nice operations the question becomes tractable.

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  • $\begingroup$ This is a cool question, it offers a lot of room for generalization by replacing associativity with any functional equations one likes. And by asking about analytic functions the same Disk metric naturally lifts up there as well. $\endgroup$
    – Sidharth Ghoshal
    Oct 31 at 2:35

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