On the plane there are several points, all the pairwise distance between them are different. Each of these points connect with the nearest. Can this set be a closed broken line?
Suppose that we have closed polygon. Let AB be the largest element of the broken line, AC and BD be an adjacent links. Then AC < AB, i.e. B is not the closest to the point A, and BD < AB, i.e. A is not the closest to the point B. Hence the points A and B cannot be connected. A contradiction occurs.