Regions in the nonstandard plane that contain no standard points

Nonstandard methods are applied to continuum theory in the plane. The main result concerns 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 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 noninfinitesimal distance from the arc joining the other two ends. 2010 Mathematics Subject Classification 54J05, 54F15 (primary); 03H05 (sec- ondary)