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
