For (2), let Y denote the set of rational points of the interval [–1,0]*1. Let S denote the union of all line segments joining the point q=(0,0) to the points of Y. Now with T as described in the question, take . Then R is path-connected, because every point in R is on a path to the origin, either in a direct line (if the point is in S) or by going along a line to p and then down the y-axis to the origin (if the point is in T). But R is not locally connected anywhere.