# A set of points on a sphere

I found this interesting question, and I was wondering if anyone could help me out.

Let P be the set of points M on the earth with the property that if you go 7 miles North from M, then 7 miles West, and finally 7 miles South, you will find yourself back at the starting point M. Is P a closed set? If not, what is the closure of P?

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The “obvious” solution is the South Pole $S$. If you travel 7 miles north from $S$, it doesn’t matter how much you travel west, you’re still going to get back to $S$ once you go 7 miles south again.

The less obvious solutions are those points that are further than 7 miles from the north pole, and are such that after traveling 7 miles north, you lie on a latitude such that the circumference of that latitude is equal to $\frac{7}{n}$ miles, for a natural number $n$. If you travel 7 miles north (again, assuming that you can at all), and reach a point such that traveling 7 miles west has no ultimate effect, you will get back to where you started after going 7 miles south again.

As Mariano points out below, it would not even mean anything to “travel 7 miles north” if you are less than 7 miles from the north pole, so we must exclude these points explicitly to make sure the condition is specified for all points.

In other words, if $A_n$ is the latitude with circumference $\frac{7}{n}$, and $B_n$ is the set of points which, after traveling 7 miles north, you would reach $A_n$, then the points on $B_n$ are solutions.

$$P=\{S\}\cup\bigcup_{n\in\mathbb{N}}B_n$$

This is not a closed set; to produce the closure, you have to add the points that are exactly 7 miles below the North Pole – this corresponds to a circumference of 0, or “$B_\infty$”. That is,
$$\overline{P}=\{S\}\cup\left(\bigcup_{n\in\mathbb{N}}B_n\right)\cup B_\infty$$

Here is an extremely not-to-scale drawing: