Dense subsets in the hyperspace of compact sets

Question

Let $X$ be compact and Polish (I'm thinking of $X=[0,1]$, but I guess maybe the same answer holds for every compact Polish space) and let $\mathbf{K}(X)$ be the hyperspace of compact subsets of $X$, endowed with the Vietoris topology. Recall that a prebase for the Vietoris is made of sets of the type $\{K\in \mathbf{K}(X):K\subset U\}$ and $\{K\in \mathbf{K}(X):K\cap U\neq\emptyset\}$ for $U\subset X$ open.

I am wondering how a dense $G_\delta$ subset of $\mathbf{K}(X)$ should look like. It is easy to think of sufficient conditions for a set to be a dense $G_\delta$: let $U_n$ be open and dense in $X$, and define $\mathcal{U}_n=\{K\in \mathbf{K}(X):K\subset U_n\}$. The set $\mathcal{U}_n$ is dense in $\mathbf{K}(X)$ and therefore so is $\bigcap_n \mathcal{U}_n$. But of course not all dense $G_\delta$ sets are of this type.

I bet there is something in the literature about this. Can somebody point me to some work where this question has been answered?

Just an example of a dense $G_\delta$ set not of that form, and also an example which shows that the answer might depend on $X$. Let $K_p(X)={K\in K(X)\mid K\text{ is perfect}}$, then $K_p(X)$ is $G_\delta$ and if $X$ is perfect, as in the $X=[0,1]$ case then $K_p(X)$ is dense in $K(X)$, but it isn't in general. — Alessandro Codenotti, Jun 05 '20 at 14:49
My comments to the question The collection of all compact perfect subsets is $G_\delta$ in the hyperspace of all compact subsets will lead you to many examples in which certain collections of small sets have first (Baire) category complements in $\mathbf{K}([0,1]),$ and most of those collections wind up actually being $G_{\delta}$ subsets of $\mathbf{K}([0,1])$ (rather than just being a superset of a dense $G_{\delta}$ subset). See also this google search. — Dave L. Renfro, Jun 05 '20 at 15:56

Dense subsets in the hyperspace of compact sets

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