A theorem in topology says:

For every continuous map there exists a pair of anitpodal points and in such that .

Specializing to the case , one might conclude that at any point of time there are two antipodal points on Earth’s surface (which is homeomorphic to ) having same, say, pressure and temperature (which together constitute of the theorem) .

Question: Isn’t it that the presence of polar caps (The Arctic and The Antarctica) where the day/night variation in temperature is quite low, may then be looked upon as a kind of `consequence’ of this theorem? (ofcourse the theorem doesn’t say where is located but at almost all other regions on Earth’s surface, the antipodal points are expected to have quite different temperatures due to day/night variation.)

ps1: One might worry about the validity of the assumption of continuity of and since there are wild local fluctuations; but I feel once you coarse-grain things out, this is a reasonable assumption.

ps2: For proof of the theorem, see http://www.mi.ras.ru/~scepin/elem-proof-reduct.pdf

Actually, I think you can make the statement stronger(This is a particular application of Borsuk-Ulam theorem) .

But as shown, for example here and here, a much stronger statement follows from simple application of Intermediate value theorem – Given a continuous function over the sphere and

any given great circledrawn on the sphere, there exists an antipodal pair of points lying on that great circle, where the function takes the same value.This in turn means that you can find such a set of points over any great circle – for example, every longitude should have such points and so should the equator. That being the case, I don’t think your statement that

is true.

P.S. : You might find the debate at this blog interesting. Especially, read the comments !

I realize flaw in my argument. Thanks for the links.

I just chanced upon this post and its comments, and am slightly concerned about the possibility of people thinking (wrongly)that the Intermediate Value Theorem proves the Borsuk-Ulam theorem (even for n=2) or a ‘stronger’ result. It does not. The proof of the Borsuk-Ulam theorem uses some algebraic topology.

Re : Subhojoy’s comment – I am sorry for the vague language in my comment above.

What Intermediate value theorem proves(correct me if I am wrong) is Borsuk-Ulam for n=1. Since we are talking about a single function (Temperature), (n=1) Borsuk-Ulam applies to temperature on a great circle. And the intermediate value theorem is enough to prove it.

I don’t know of a simple proof of Borsuk-Ulam for n greater than or equal to 2, of course (and it might very well be necessary to invoke algebraic topology).We need n=2 Borsuk-Ulam, only if we want to assert that there is an antipodal pair on earth surface where pressure

andtemperature are the same. But, until we seek to assert such a strong statement abouttwofunctions, we can get away with intermediate value theorem.I hope that clarifies the confusion.

Yes, that is correct. In which case, it still remains an interesting ‘geography’ question as to which two antipodal points do have the same P and T, and how that pair evolves with time.