An example of a rigid κ-superuniversal metric space

For a cardinal κ>ω a metric space X is called 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 [2] and Katětov [6]. 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⊆K with |Y|<κ can be extended to an isometry of the whole K. In connection to this it has been suggested [W. Kubiś, personal communication, 2012] that there should also exist a κ-superuniversal space that is not κ-homogeneous. In this paper it 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.