Ambiently universal sets in 𝐸ⁿ

Abstract

For each closed set X X in E n {E^n} of dimension at most n − 3 n - 3 , we show that X X fails to be ambiently universal with respect to Cantor sets in E n {E^n} ; i.e., we find a Cantor set Y Y in E n {E^n} so that for any self-homeomorphism h h of E n {E^n} , h ( Y ) h(Y) is not contained in X X . This result answers a question posed by H. G. Bothe and completes the understanding of ambiently universal sets in E n {E^n} .