Externalities, Nonconvexities, and Fixed Points

But expression (66) contradicts expression (61) stating that for all n ≥ N 1 ,ρ Z (z 00 ,z 0 )+ρ w ∗(f k n(z 00 ),l 0 ) > η 0 for all z 00 ∈ B δ n(z n ).Therefore, we must conclude thatF ∗ (z) =κ(η(z)) := ∩ {M ∈ C w ∗ f(X) :η(z) ⊆ M} for all z ∈ Z,implying that for all z ∈ Z, F ∗ (z) ∈ C w ∗ f(X) is the unique subcontinuum irreducibleabout η(z) ∈ P w ∗ f(X).It remains to show thatF ∗ (z) ⊆ N (z) for all z ∈ Z.To begin, define the limit point mapping L(·) as follows:L(z) :={x ∈ F ∗ (z) :(z,x) is a limit point of F ∗ \{(z,F ∗ (z))}} .L(·) is an USCO (i.e., L(·) ∈ U(Z, P w ∗ f(X))). Because F ∗ (·) is usc with connectedvalues, it follows from Hiriart-Urruty (1985) that F ∗ (= GrF ∗ (·)) is connected.Thus, GrL contains no isolated points, implying that L(·), in addition tobeing usc, automatically satisfies conditions (b-1) and (b-2) of Beer’s dense selectionTheorem above, further implying via Beer’s Theorem 4 (1983) that L(·) has adense selection, g ∗ . 17 But now by the Corollary 6 in Crannell, Franz, and LeMasurier(2005), because f ∗ (z) =g ∗ (z) for all z in the ρ Z -dense open G δ set D in Z where{f ∗ (z)} = η(z) =F ∗ (z),Grη = Grf ∗ = Grg ∗ = GrL.Because F ∗ has no isolated points, it follows from Beer’s Theorem on dense selectionsthat if (z 0 ,x 0 ) ∈ © (z 0 ,F ∗ (z 0 )) ª ©, then either (z 0 ,x 0 ) is a limit point of F ∗ \(z 0 ,F ∗ (z 0 )) ª or (z 0 ,x 0 ) is a limit point of © (z 0 ,F ∗ (z 0 )) ª only. Given the definitionof the limit point mapping, L(·), (z 0 ,x 0 ) is a limit points of F ∗ \ © (z 0 ,F ∗ (z 0 )) ª ifandonlyifx 0 ∈ L(z 0 ),and(z 0 ,x 0 ) is a limit point of © (z 0 ,F ∗ (z 0 )) ª only if andonly if x 0 ∈ F ∗ (z 0 )\L(z 0 ). Moreover, if x 0 ∈ F ∗ (z 0 )\L(z 0 ),thenz 0 ∈ X\D. Nowsuppose that for some z 0 ∈ Z, thereisapointx 0 ∈ F ∗ (z 0 ) not contained in N (z 0 ).Because η ∈ [N ] ρZ -w ∗ and Grη = GrL and because x0 ∈ F ∗ (z 0 )\L(z 0 ), implies thatz 0 ∈ X\D, F ∗ (z 0 ) has at least two values. Thus, we know by strong convexity thatthere is a unique segment x 1 x 2 ⊂ F ∗ (z 0 ) containing x 0 - and we have by argumentsgiven in the proof of the first part of (2) above that x 1 ∈ L(z 0 ) and x 2 ∈ L(z 0 ).Because F ∗ (z 0 ) is a dendrite and because x 0 ∈ x 1 x 2 ,notequaltox 1 or x 2 , we knowby Whyburn (1942, 1.1) (also see part (3) of Theorem 1.1 in Charatonik and Charatonik,1998) that x 0 is a cut point of F ∗ (z 0 ) (rather than an endpoint of F ∗ (z 0 ) -by part (3) of Theorem 1.1 in Charatonik and Charatonik, 1998, because F ∗ (z 0 ) isa dendrite, x 0 ∈ F ∗ (z 0 ) is either an endpoint or a cut point). Let {x 0 ,V 1 ,V 2 } be17 Recall that in a topological space a point z is isolated if {z} ∩ U z = {z} for all neighborhoodsU z of z. Thepointz is a limit point if for each neighborhood U z of z contains a point z 0 (6= z).37