Do infinitely many points on earth have the same temperature as their antipodal?

Let $$X=S^2$$ be the unit sphere in $$\mathbb{R}^3$$ and $$T:X\rightarrow \mathbb{R}$$ be a continuous function.

My topology textbook claims that the set $$A=\{x \in X\ |\ T(x)=T(-x)\}$$ has an infinite number of elements.

The fact that $$A$$ is non empty is clear to me as a consequence of the intermediate-value theorem, since $$f:X\rightarrow \mathbb{R},\ x \mapsto T(x)-T(-x)$$ is continuous, X is connected and $$f(X)$$ contains a non-positive and a non-negative real number.

What's way less clear is how there can't be a finite number of points in $$A$$. My intuition is that there must be a (non-trivial) curve on the sphere that contains the antipodal of each of its points, but I really don't know how to show it, if that's even true.

• Hint: consider the great circles! Jul 24, 2020 at 10:21
• The title is misleading. Jul 24, 2020 at 10:22
• @Mindlack I thought this had something to do with the great circles, but I can't find a way to write them as subsets of the sphere, so I don't know how to formalize what's in my head. Jul 24, 2020 at 10:36
• @AsafKaragila I'm modelling the surface of the earth as the unit sphere and the temperature as a continuous function from the unit sphere to the reals. Would it be less misleading if I wrote that in the question? Jul 24, 2020 at 10:38

Take a point $$p \in X \setminus A$$ and call it northpole, call $$-p$$ southpole. Now we look at the longitudes, the circles (longitudes) through these poles on the sphere.

Take one longitude and call it $$L$$.

$$f$$ is defined on $$L$$ and it is nonzero on $$p$$. As $$f(-p) = -f(p)$$ we know that $$f$$ has both negative and positive points on $$L$$. As $$L$$ is connected and $$f$$ is continous this means that there is a point $$z_L$$ on $$L$$ where $$f (z_L)=0$$. So $$z_L \in L \cap A$$. $$p$$ and $$-p$$ are not in $$L \cap A$$, though, so $$z_L$$ is not one of the poles.

As this is true for each of the infinite longitudes $$L$$ this means that there is a point in $$A$$ for every longitude. These points are distinct as the longitudes only meet in the poles (which are not in $$A$$). Therefore $$A$$ must have at least as many points as there are longitudes, so $$A$$ is inifinite.

qed

(Slightly away from your specific question, if you are working on something as similar to the Borsuk-Ulam theorem then you can't do yourself a bigger favour then watching this most excelent video from 3b1b about it: https://www.youtube.com/watch?v=yuVqxCSsE7c)

• You're making the implicit assumption that $X\setminus A\ne\emptyset$. Of course if that is not the case, then $A$ quite obviously has infinitely many points as well. Jul 24, 2020 at 10:59
• @celtschk You are right. For a formal proof you'd have to mention that I guess. Jul 24, 2020 at 11:01
• 3b1b is always one of the best sources to get the intuition behind very formal ideas! Jul 24, 2020 at 11:07