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

Question

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.

Answer 1

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)