Lusternik–Schnirelmann theorem

Remarks

The theorem follows from the Borsuk–Ulam theorem. We may assume that every $A_i$ is nonempty, by replacing an empty one with a copy of a nonempty one. Let $f(x) = (d(x, A_1), \ldots, d(x, A_n)) \in \R^n$, where $d(x, A)$ is the distance from $x$ to $A$, and choose $x$ with $f(x) = f(-x)$. If some coordinate of $f(x)$ is $0$, then $x$ and $-x$ both lie in the closed set $A_i$; otherwise neither lies in $A_1 \cup \cdots \cup A_n$, so both lie in $A_{n+1}$. The same conclusion holds when all the $A_i$ are open.

Example: for $n = 1$, if the circle is the union of two closed sets, one of them contains two opposite points.

The number $n + 1$ cannot be raised to $n + 2$. Project the facets of a regular $(n+1)$-simplex centred at the origin radially onto $S^n$. This covers $S^n$ by $n + 2$ closed sets, none of which contains an antipodal pair.

Going deeper

For a student

The theorem is one of several equivalent forms of the Borsuk–Ulam theorem, and it can be proved from it and conversely. It gives a short proof of Kneser's conjecture on the chromatic number of Kneser graphs. More generally, covering and coincidence statements of this kind are a main entry point of topological methods in combinatorics.