Borsuk–Ulam theorem for $n=2$

Question

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures.

What about a rigorous proof?

Answer 1

Assume the temperature map $T (x,y)$ , i.e. the assignment of temperature at different points in $S^n$ is a continuous map.

Then you can use the map: $$h: (x_1,x_2) \rightarrow (T(x_1,x_2),T(-x_1,-x_2))$$ , which is a continuous map from $S^2$ into $\mathbb R^2 $. Then, by Borsuk-Ulam, there are $(x_i,y_i),(-x_i,-y_i)$ in $S^2$ with $h(x_i,y_i)=(T(x_i,y_i),T(-x_i,-y_i))= h(-x_i,-y_i)=(T(-x_i,-y_i), T(x_i,y_i))$. It follows that $T(x_i,y_i)=T(-x_i,-y_i)$.