An example of a rigid \(\kappa\)-superuniversal metric space (Q290624)

For a cardinal \(\kappa >\omega\) a metric space \(X\) is said to be \(\kappa\)-superuniversal whenever for every metric space \(Y\) with \(| Y| <\kappa\) each partial isometry from a subset of \(Y\) into \(X\) can be extended over the whole space \(Y.\) Such spaces were studied by several authors. For instance Katětov established that if \(\omega<\kappa=\kappa^{<\kappa},\) then there exists a \(\kappa\)-superuniversal space \(K\) which is also \(\kappa\)-homogeneous, that is, every isometry of a subspace \(Y\subseteq K\) with \(| Y| <\kappa\) can be extended to an isometry of the whole space \(K\).NEWLINENEWLINEIn the present paper the author studies a problem suggested by W. Kubiś looking for \(\kappa\)-superuniversal spaces that are not \(\kappa\)-homogeneous.NEWLINENEWLINEThe main result states that for every cardinal \(\kappa\) there exists a \(\kappa\)-superuniversal space which is rigid, that is, possesses exactly one isometry, namely the identity.NEWLINENEWLINEThe described construction is based on a detailed study of an amalgamation-like property of a family of metric spaces.