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**.

=====================================================

 

    **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*