Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris topology. ¿Has it been proven that the $n$-fold deleted symmetric product $[X]^{n} := \{A\subseteq X : |A| = n\}$ is also connected when equipped with the Vietoris topology? A colleague and I are under the impression that someone already did this, but we haven't found a reference to confirm it.