On ambient embeddings in the Hilbert cube (Q1083072)

The ambiently universal subsets of a given space S are the ones large enough to absorb all members from another prescribed family of subsets under self-homeomorphisms of S. Building on work of H. G. Bothe, who extensively studied various kinds of ambiently universal sets in Euclidean n-space \(E^ n\), the author showed in an earlier paper [Trans. Am. Math. Soc. 277, 655-664 (1983; Zbl 0531.57014)] that no closed subset of \(E^ n\) having dimension less than n-2 can be ambiently universal with respect to Cantor sets. This means that corresponding to every (n- 3)-dimensional closed subset X of \(E^ n\) there exists a Cantor set K in \(E^ n\) for which no self-homeomorphism h of \(E^ n\) carries K into X. Here he derives essentially the same thing for the Hilbert cube Q, proving that no codimension three compact subset of Q is ambiently universal with respect to Cantor sets.