Rim-finite, arc-free sets
by
John Kulesza
George Mason University
Coauthors: Jay Schweig

A topological space is rim-finite (rim-n) if it has a basis of open sets whose boundaries are finite (contain at most n points). A subset of R² is a partial n-point set if it contains at most n points of each straight line, and is an n-point set if it contains exactly n points of each straight line. A set X is a GM set in R² if it contains exactly two points of any line which separates points of X. All partial n-point sets, n-point sets and GM sets are rim-finite.

We prove somewhat more general versions of this theorem:

Theorem. If X subset R² is such that there are dense subsets H and V of the horizontal and vertical line sets whose elements all contain at most n points of X, then either X contains an arc or X is zero-dimensional.

As a corollary to this theorem, all arc-free planar sets which are partial n-point, n-point, or GM are zero-dimensional. This answers questions of Bouhjar and Dijkstra, and also Loveland and Loveland.

An example is provided of a rim-72 arc-free subset of R² which is not zero-dimensional. This example illustrates limitations for generalizing the theorem, and can be modified to be nonplanar.

Date received: May 15, 2001

Copyright © 2001 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 # cagw-44.