The paper Social choice and topology a case of pure and applied mathematics by Beno Eckmann investigates what is meant by a mean on a topological space. In particular, if $k$ is a natural number, then they define a $k$-mean to be a continuous function $f:X^{k}\rightarrow X$ such that $f(x,...,x)=x$ and $f(x_{1},...,x_{k})=f(x_{\sigma(1)},...,x_{\sigma(k)})$ where $\sigma:\{1,...,k\}\rightarrow\{1,...,k\}$ is any permutation. (I should mention that I found out about this paper from this question)
Joseph Van Name
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