There is this old chestnut: "A man walks a mile South, a mile East, and a mile North and ends up right back where he started. Where did he start?"
Aside from the timeworn singular point answer, give another answer that includes a countably infinite number of sets, each containing an uncountably infinite number of points that satisfy the problem. (Assume a smooth spherical globe, and no tricks.)

The North Pole is the obvious answer.
But one can also start a little over one mile away from the South Pole, walk one mile South, and arrive at a point at a distance R from the South Pole.  At this point walking East you travel 1 mile = 2 pi R distance to arrive at your starting point.  Then walk one mile North.
So there are an infinity of points that are located a distance 1 + R from the South Pole:  anywhere along that line of latitude.
Not only that, but during your circular journey around the South pole, you could do any integral number of "laps" and have the same result.  So there are an infinity of latitudes you can start at.

1 mile = 2 n pi R
R = 1 / (2*n*pi) miles  {n = 1, 2, 3, ...}
Starting points:  North Pole or anywhere a distance of 1 + R from the South Pole

Posted by Larry on 2018-07-02 08:48:51