An example of a rigid $κ$-superuniversal metric space

For a cardinal $κ> ω$ a metric space $X$ is called to be $κ$-superuniversal whenever for every metric space $Y$ with $|Y| < κ$ 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 Katětov [2]. In particular, Katětov showed that if $ω< κ= κ^{< κ}$, then there exists a $κ$-superuniversal $K$ which is moreover $κ$-homogeneous, i.e. every isometry of a subspace $Y\subseteq K$ with $|Y|<κ$ can be extended to an isometry of the whole $K$. In connection of this W. Kubiś suggested that there should also exist a $κ$-superuniversal space that is not $κ$-homogeneous. In this paper there is shown that for every cardinal $κ$ there exists a $κ$-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.

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