Let $X$ be a topological space. Let $[X]^{<\omega}$ be the space of non-empty finite subsets of $X$, endowed with the Vietoris topology. For a natural number $n$ the subspace $$[X]^{\le n}:=\{A\in[X]^{<\omega}:0<|A|\le n\}$$ of $[X]^{<\omega}$ is called the $n$-th hypersymmetric power of $X$.

It is easy to see that the map $X^n\to [X]^{\le n}$, $(x_1,\dots,x_n)\mapsto\{x_1,\dots,x_n\}$, is perfect. Consequently, the normality of $X^n$ implies the normality of $[X]^{\le n}$. What about the inverse implication?

Problem. Is there a topological space $X$ such that for some $n\in\mathbb N$ the hypersymmetric power $[X]^{\le n}$ is normal but the power $X^n$ is not?

Remark. In Asymptology there exists an example of a coarse space $X$ whose square $X^2$ is not normal but all hypersymmetric powers $X^{\le n}$ are normal.


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