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\subseteq U\ $ in $\ \mathscr U\ $ such that space $\ (C\,\ \delta|C\!\times\!C)\ $ is universal for all finite metric spaces?

Similar questions 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.

PS. Mathematicians, please, be tolerant! Do not edit my STYLE.

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    their story  

    she talked talked talked
    then married
    his confidence
    his cocksure ways

    she kept talk-talk-talking
    for the next twenty years
    till she broke down
    that damn confidence of his
    that cocky style

    these days
    she plays mean darts
    drinks beer
    talk-talk-talks to poor bastards for miles
    searching for the brimming confidence
    for the cocksure smile

Włodzimierz Holsztyński -- 1998-02-12