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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.

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    $\begingroup$ Hint: consider the great circles! $\endgroup$
    – Aphelli
    Jul 24, 2020 at 10:21
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    $\begingroup$ The title is misleading. $\endgroup$
    – Asaf Karagila
    Jul 24, 2020 at 10:22
  • $\begingroup$ @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. $\endgroup$ Jul 24, 2020 at 10:36
  • $\begingroup$ @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? $\endgroup$ Jul 24, 2020 at 10:38

1 Answer 1

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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)

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    $\begingroup$ 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. $\endgroup$
    – celtschk
    Jul 24, 2020 at 10:59
  • $\begingroup$ @celtschk You are right. For a formal proof you'd have to mention that I guess. $\endgroup$
    – Kaligule
    Jul 24, 2020 at 11:01
  • $\begingroup$ 3b1b is always one of the best sources to get the intuition behind very formal ideas! $\endgroup$ Jul 24, 2020 at 11:07

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