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Wlod AA
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universal 0-dimensional separable metric subspaces

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.

Wlod AA
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