Let $\ \mathscr U:=(U,\delta)\ $ be a separable metric space which is universal for all finite metric spaces, i.e. for every finite metric space $ \mathscr X:=(X,d)\ $ there exists an isometric embedding of $\ \mathscr X\ $ into $\ \mathscr U.$
Q: Does there exist a 0-dimensional subset $\ C\subset U\ $ in $\ \mathscr U\ $ such that space $\ (C,\delta|C\!\times\!C)\ $ is universal for all finite metric spaces?
Similar question hold for
the subcategory of the above metric spaces of diameter $\le 1;$
the subcategory of the above metric spaces which are complete.
As long as I know, these questions are open.
