On the dimension of almost n-dimensional spaces
by
E.D. Tymchatyn
University of Saskatchewan
Coauthors: M. Levin

A separable metric space X is said to be almost n-dimensional if there is a basis { U_i } such that cl (U_i ) \cap cl (U_j ) = \phi implies X = G \cup H where G and H are closed sets, U_i subset G \H, U_j subset H \G and dimG \cap H <= n - 1. The space of irrational sequences in Hilbert space is an almost 0-dimensional space which is 1-dimensional.

Theorem. If X is almost n-dimensional and n >= 1 then X is at most n-dimensional.

A space is said to be hereditarily locally connected if it is connected and each of its connected subsets is locally connected. The following corollary answers an old question of R. Duda.

Corollary. Every non-degenerate, separable, metric, hereditarily locally connected space is 1-dimensional.

Date received: March 3, 1997

Copyright © 1997 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 # caam-44.