= Open-cover proof of the Lusternik-Schnirelmann-Borsuk theorem
For a finite <open cover> $U_0,\ldots,U_d$ of $S^d$, take continuous nonnegative functions $\phi_i$ summing to one and positive only in $U_i$. Such a <partition of unity> comes from normalizing the <distance to a set> functions $d(x,S^d\setminus U_i)$. The <Borsuk-Ulam theorem> applied to $(\phi_1,\ldots,\phi_d)$ gives equal values at an <antipodal pair>. The sum condition makes $\phi_0$ equal too, and some common positive value places both points in one $U_i$.
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