# The normality of powers versus the normality hypersymmetric powers

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.