Vol. 24, No. 1, 1968

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A wild Cantor set in the Hilbert cube

Raymond Y. T. Wong

Vol. 24 (1968), No. 1, 189–193

Let En be the Euclidean n-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 En for n 3, where “wild” means that En 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 Hilbert space l2. A result of this paper provides a positive answer for a generalization of Blankinship’s result in the Hilbert cube.

Mathematical Subject Classification
Primary: 54.78
Received: 28 October 1966
Revised: 1 May 1967
Published: 1 January 1968
Raymond Y. T. Wong