On Unbounded Z-homeomorphisms in Rⁿ
by
Otto Laback
Graz University of Technology

We call a subset A of a topological R-vector space X an (image of) a finite piecewise linear arc (fpla) if there is a sequence (a_n), n=1, .., k such that every element of A is either one of the a_i or can be written as the open line segment between a_i and a_i+1 for some i=0, .., k-1, and no element not belonging to A can be written in such a way.An example (L. Rubin) of an fpla preserving transformation which is unbounded (in a proper sense, such that it is representable by an autohomeomorphism with respect to a finest topology), can be constructed by the following bijection F:R^3-> R^3 . Construct a sequence of points (a_n ) in the x-y-plane on the parabola y=x^2, x>0 which converges fast enough to (0, 0, 0). Construct further a sequence of circles around these points and parallel to the z-axis cylinders with disjoint interiors. Finally transform the space R^3 by F such that outside of the cylinders F will be the identity but inside the sequence of cylinders F tansforms the axis parallel to z ``successively towards infinity".I would like to investigate the images of certain dense lying subsets of a compact Euclidean ball (S. Popvassilev) under such unbounded transformations.The Zeeman-topology Z on R^n is defined as the finest topology which induces theEuclidean topology on each fpla. There are interesting unbounded auto-homeomorphisms with respect to the topology Z

Date received: May 19, 1999