Rim-finite, arc-free subsets of the plane

Abstract

We investigate properties of rim-finite subsets of the plane (those which have topological bases whose elements have finite boundaries), which are also arc-free. Recently (see [K. Bouhjar, J.J. Dijkstra, Preprint], [K. Bouhjar, J.J. Dijkstra, J. van Mill, Topology Appl., to appear], [M.N. Charatonik, W.J. Charatonik, Comment. Math. Univ. Carolin., to appear], [D.L. Fearnley, J.W. Lamoreaux, Proc. Amer. Math. Soc., to appear] and [L.D. Loveland, S.M. Loveland, Houston J. Math. 23 (1997) 485–497]) there has been considerable research regarding n-point sets (sets which intersect each line in exactly n-points). These spaces are rim-finite (since the interior of a triangle has its boundary contained in a union of three lines, each of which has n points of the space), and our investigation provides a direction to generalize them. One of our main theorems seems to generalize all known results regarding the dimension of n-point sets (see, for example, [K. Bouhjar, J.J. Dijkstra, J. van Mill, Topology Appl., to appear], [D.L. Fearnley, J.W. Lamoreaux, Proc. Amer. Math. Soc., to appear] and [J. Kulesza, Proc. Amer. Math. Soc. 116 (1992) 551–553]), and beyond that has, as corollaries, the solutions to problems of Bouhjar and Dijkstra [Preprint], and L.D. Loveland and S.M. Loveland [Houston J. Math. 23 (1997) 485–497]. In Bouhjar and Dijkstra [Preprint] it is asked if all n-point sets which are arc-free must be zero-dimensional, and our result gives a positive answer. In [L.D. Loveland, S.M. Loveland, Houston J. Math. 23 (1997) 485–497] it is asked whether a connected 2-GM set must contain an arc, and again we give a positive answer.Another main theorem states that if X is a subset of R2 such that there is a nonnegative integer n so that every straight interval of length 1 has a local basis of open sets with boundaries which intersect X in a set of cardinality less than or equal to n, then either X is zero-dimensional or X contains an arc. We produce an example which demonstrates that, essentially, our theorem cannot be improved. The “straight interval of length 1” cannot be replaced by “point”, because our example has a base of open sets whose boundaries have cardinality less than or equal to 72 and contains no arcs, yet has dimension 1. This example seems to be the first of a positive dimensional, rim-finite and arc-free separable metric space.