Large superuniversal metric spaces

Abstract

For every uncountable cardinal κ define a metric spaceS to be κ-superuniversal iff for every metric spaceU of cardinality κ, every partial isometry intoS from a subset ofU of cardinality less than κ can be extended to all ofU. We prove that any such space must have cardinality at least$2^{\bar \kappa } = \sum _{\lambda< \kappa } 2^\lambda$, and for each regular uncountable cardinal κ, we construct a κ-superuniversal metric space of cardinality$2^{\bar \kappa }$, It is proved that there is a unique κ-superuniversal metric space of cardinality κ iff$2^{\bar \kappa } = \kappa$. Several decomposition theorems are also proved, e.g., every κ-superuniversal space contains a family of$2^{\bar \kappa }$ disjoint κ-superuniversal subspaces. Finally, we consider some applications to more general topological spaces, to graph theory, and to category theory, and we conclude with a list of open problems.