    Convex Klee (Proposition 3) ⇄ Convex KKM (Theorem 6) ⇄ FPT for N (Theorem 9) ↑ ↓ ? ↖ ↘ ? ↑ ↓ ? ? ↙ ↗ ↑ ↓ ? KKM Lemma ⇄ Ky Fan KKM ⇄ Browder-Ky Fan FPT     Let us point out, for the reader’s benefit that the following equivalences have been established in [15]:   $^{Brouwer FPT}$ ⇄ Topological Klee ↖↘ ↙↗ Ky Fan KKM    Here, the ”topological Klee theorem” [15] reads: A family of n closed convex sets in a topological vector space has a non- empty intersection if and only if the union of the n sets is ( n − 2) -connected and the intersection of every n − 1 of them is non-empty. This topological version of Klee’s theorem yields the equivalent formulation of the Brouwer fixed point theorem: The n -sphere S n is not n -connected. Indeed, the n − dimensional faces of the ( n +1) − simplex Δ n +1 form a family of n + 2 closed convex sets in R n $^{+2}$. Moreover, every intersection of n + 1 of them is non-empty, but the whole intersection is empty. Hence, their union - which consists of the boundary ∂ Δ n +1 - is not n − connected. Since ∂ Δ n +1 is homeomorphic to S $^{n}$, it follows that S n is not n − connected. This establishes 21