A wild Cantor set in the Hilbert cube

Abstract

Let E n be the Euclidean w-space. A Cantor set C is a set homeomorphic with the Cantor middle-third set. Antoine and Blankinship have shown that there exists a "wild" Cantor set in any E n for n ^ 3, where "wild" means that E n -C is not simply connected. However it is also known that no "wild" Cantor set (in fact, compact set) can exist in many infinite dimensional spaces, such as s (the countably infinite product of lines) or the Hubert space l 2 . A result of this paper provides a positive answer for a generalization of Blankinship's result in the Hubert cube.