Towards Chainability For Certain Regions Obtained From Simple Dense Canals

Nonstandard methods are used to obtain results about sets formed by simple dense canals in the plane. The …rst main result considers internal regions V that contain no standard points and have the property that st(V ) does not disconnect the plane and are bounded by simple closed curves that have small intersection with st(V ): We show that such sets must be somewhat one-dimensional in the sense that they cannot contain a “Y-set” consisting of three arcs intersecting only at a single common point with the property that the non-intersecting endpoints of each arc are a nonin…nitesimal distance from the arc joining the other two ends. The second main result is that such sets can always be formed by “closing o¤”sections of simple dense canals with arcs of in…nitesimal length in such a way that the closed o¤ portion passes within an in…nitesimal distance of every point of the entire canal. The relevance of these results and techniques to the plane …xed point problem is discussed. Key words: nonstandard topology, …xed points