... acompact subset of C, then f has a fixed point in C.Proof. The closed convex hull K= cl-co f (C) of the set f (C)isacompactconvexsubset of C.Since f (K)⊂ f (C) ⊂ K,then,byTheorem 2.3, the mapping ... F(xγ( j))andyγ( j)→ y, so that, by the closedness of F, y∈ F(x). By the choice of the elements yi, the elements yγ( j)belong to the closed setY\ V,aswellastheirlimity,implyingy ∈ F(x) \ ... B1.IfxB∈ X, B ∈ Ꮾ,then(xB: B ∈ Ꮾ(x)) is a net in X. We denote by ᐂ(x) the family of all neighborhoods of apoint x∈ X,andbycl(Z)theclosureofasubsetZ of X.We will use the following facts.Proposition...