A Regular Space with a Countable Network and Different Dimensions
by
Stephen Watson
York University
Coauthors: George Delistathis

In the study of dimension, a fundamental theorem states that for separable metric spaces, the inductive dimensions suggested by Poincare coincide with the covering dimension of Lebesgue. It was natural to attempt to extend this coincidence to larger classes of spaces. P. Roy showed in 1965 that this coincidence did not hold for arbitrary metric spaces. Arhangelskii had introduced the notion of network in 1959 to handle spaces which are unions of a small number of separable metrizable subspaces, and so it was natural and central to ask whether the coincidence of dimension could be established for spaces with a countable network. At the International Congress of Mathematicians in Nice in 1970, a paper of Arhangel'skii was presented in which this question was asked. Some partial results were established by Leibo and Oka during the 1970's and related results were obtained by Filippov and the Georgian school of Zambahidze but the main problem has remained open for nearly 30 years.

We construct a regular space X with a countable network such that dim(X) = 1 and ind(X) = 2. We discuss whether the continuum hypothesis is needed for our construction and whether the gap between the dimensions can be widened. The space is a subspace of a resolution of the unit square. The methods in our proof include viewing a resolution as a transfinite inverse limit and then iterating a dimension-raising process on Cantor sets which is based on Kuratowski's one-dimensional graph over the Cantor set. A major difficulty was created by our need to preserve this dimension-raising process while maintaining a certain countable network. We state several fascinating and related problems which remain open.

Date received: February 1, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caab-77.