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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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    If you are 7 miles South from a polar circle whose circumference is 7 miles then you will arrive back at your starting point. You can extend this. Suppose that you are 7 miles South from a polar circle whose circumference is 7/n miles where $n$ is an integer, you can go 7 miles North to this circle, travel $n$ times around and go back 7 miles South. I'm not sure what the closure will be.2011-11-11
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    How do you go 7 miles North from the North pole?2011-11-11
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    @AleksVlasev has correctly described the set of such points (with the exception of the South Pole itself).2011-11-11

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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:

enter image description here

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    Also, I think we should note that $B_{\infty}$ itself is not a solution because there is no notion of "traveling 7 miles west" when you are at the North pole.2011-11-11
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    @Aleks: I think my answer makes it clear that $B_\infty$ isn't a solution, but I will try to make it clearer :)2011-11-11
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    I would say that the definition of the set $P$ simply does not make sense, in that for points no further away from the North pole than 7 miles the condition for a point to belong to $P$ is meaningless —there is no sense in which one can say that they satisfy or not the condition, for the condition is meaningless for them.2011-11-11
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    @Mariano: That's a good point, I've clarified my definition of $P$.2011-11-11
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    Here's a follow up question that I have in mind that I'm not sure if I should post as a separate question. What would happen if "Earth" was a small ball, say, less than 7 miles in circumference. How would the solution change?2011-11-11
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    @Aleks: Good question! Note Mariano's objection above, that traveling 7 miles north makes no sense if you are within 7 miles of the north pole. So if the Earth were less than 7 miles in circumference, it would be impossible to travel 7 miles north from any point, and so there would be no solutions. I'm also reasonably sure that we'll never have to consider the southern latitudes of radius $\frac{7}{n}$; I think, regardless of the radius of the Earth, they can't be reached by going 7 miles north. However I can't come up with a convincing argument for this at the moment.2011-11-11
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    @Zev That makes sense. However, we can play around with what "going 7 miles North actually means". One could argue that we are just starting in the Northern direction and we keep going this way regardless of where we end up. Say we start at 3 miles south of the North pole. Then going 7 miles north could constitute passing the pole and arriving 4 miles south of it.2011-11-11
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    In this framework, we can picture a planet with great circle length of 7 miles and start at the equator. Traveling 7 miles North would bring us back to the equator and traveling 7 miles south from that new point will bring us back where we started. (we've traveled around on a great circle)2011-11-11
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    Thank you very much Zev! Your reasoning and diagram made perfect sense to me!2011-11-12