An example of a rigid $\kappa$-superuniversal metric space

Abstract

For a cardinal $\kappa > \omega$ a metric space $X$ is called to be $\kappa$-superuniversal whenever for every metric space $Y$ with $|Y| < \kappa$ every partial isometry from a subset of $Y$ into $X$ can be extended over the whole space $Y$. Examples of such spaces were given by Hechler [1] and Kat\v{e}tov [2]. In particular, Kat\v{e}tov showed that if $\omega < \kappa = \kappa^{< \kappa}$, then there exists a $\kappa$-superuniversal $K$ which is moreover $\kappa$-homogeneous, i.e. every isometry of a subspace $Y\subseteq K$ with $|Y|<\kappa$ can be extended to an isometry of the whole $K$. In connection of this W. Kubi\'s suggested that there should also exist a $\kappa$-superuniversal space that is not $\kappa$-homogeneous. In this paper there is shown that for every cardinal $\kappa$ there exists a $\kappa$-superuniversal space which is rigid, i.e. has exactly one isometry, namely the identity. The construction involves an amalgamation-like property of a family of metric spaces.