Ky Fan’s theorem states that a set-valued map \(A\) defined on a nonempty compact convex subset \(X\) of a topological vector space with values in the nonempty convex subsets of \(X\) has a fixed point provided the preimage of each point is open in \(X\). The present author shows that it suffices to assume that \(X\) is star-shaped instead of convex.