Currently I am stuck in the following
Question. Let $A_1,\cdots, A_{n+1}\subset S^n$ be closed subsets such that $A_1\cup\cdots\cup A_{n+1}=S^n$. Then there is some $A_i$ that contains antipodal points.
This is a proposition (but left unproved) in a lecture of algebraic topology, in the chapter of Borsuk-Ulam theorem. However I cannot see how it is related to Borsuk-Ulam theorem. I guess this could be proved by induction, since for the case of $n=1$, neither of $A_i$ containing antipodal points contradicts the connectedness of $S^1$. If for the case of $n$, we can show that there is some equator that does not intersect with $A_{n+1}$, then the induction hypothesis can be applied to obtain the conclusion. But I cannot show the existence of such an equator, nor can I see how the Borsuk-Ulam theorem is applied, so I would like to ask how to finish the proof.
Thanks in advance...
This is the Lusternik–Schnirelmann theorem.
Here is a textbook proof. Define $f:S^n\to\Bbb R^n$ by taking the $j$-th component of $f(x)$ as the distance from $x$ to $A_j$. By Borsuk-Ulam, there is $x$ with $f(x)=f(-x)$. If the $j$-th coordinate of $f(x)$ is zero, both $x$ and $-x$ are in $A_j$. Otherwise both $x$ and $-x$ are in $A_{n+1}$.