Externalities, Nonconvexities, and Fixed Points

9.2 Characterizing Minimal USCOsWe will need the notion of a quasi-continuous function in order to fully characterizeminimal USCOs.Definition 8 (ρ Z -w ∗ -Continuity and ρ Z -w ∗ -Quasi-Continuity)(1) (ρ Z -w ∗ -Continuity) f : Z → X is said to be ρ Z -w ∗ -continuous at z ∈ Z if forevery w ∗ -open subset G of X such that f(z) ∈ G there is a ρ Z -open set U z containingz such that f(U z ) ⊂ G. The function f is ρ Z -w ∗ -continuous if it is ρ Z -w ∗ -continuousat every z ∈ Z.(2) (ρ Z -w ∗ -Quasicontinuity) f : Z → X is said to be ρ Z -w ∗ -quasicontinuous atz ∈ Z if for every w ∗ -open subset G of X such that f(z) ∈ G and for every ρ Z -openset U z containing z there exists another ρ Z -open set W ⊂ U z such that f(W ) ⊂ G.The function f is ρ Z -w ∗ -quasi-continuous if it is ρ Z -w ∗ -quasicontinuous at everyz ∈ Z.We will denote by C ρZ -w ∗ := C ρ Z -w ∗(Z, X) the collection of all ρ Z-w ∗ -continuousfunctions defined on Z with values in X, and we will denote byQC ρZ -w ∗ := QC ρ Z -w∗(Z, X)the collection of all ρ Z -w ∗ -quasicontinuous functions defined on Z with values in X.Let F := F(Z, X) denote the collection of all functions defined on Z taking valuesin X. Recall that a function, f : Z → X, is a selection from USCO Γ ∈ U ρZ -w ∗ iff(z) ∈ Γ(z) for all z ∈ Z. Denote the collection of all selections of Γ by Σ Γ . Wesay that f ∈ Σ Γ is a ρ Z × w ∗ -dense selection of Γ if Gr ρZ ×w∗f = GrΓ (i.e., if theρ Z × w ∗ -graph closure of f is equal to the graph of Γ). When no confusion can occur,we will write Grf rather than Gr ρZ ×w ∗f.The following two part Lemma is due to Crannell, Franz, and LeMasurier (2005)and Hola and Holy (2009). In our statement of these results we take as given thefact that Z and X are compact metric spaces, equipped with convex metrics, ρ Z andρ w ∗,whereρ Z is compatible with the d Z metric topology on Z and ρ w ∗ is compatiblewith the weak star topology on X. In fact, Part (1) due to Crannell, Franz, andLeMasurier (2005) requires that Z and X be compact metric spaces.Lemma 19 (Characterizing Minimal USCOs)(1) (Crannell, Franz, and LeMasurier, 2005)For all g ∈ F there exists f ∈ QC ρZ -w ∗such that Grf ⊂ Gr ρ Z ×w ∗g.(2) (Hola-Holy, 2009)The following statements are equivalent.(i) Λ ∈ M ρZ -w ∗.(ii) There exists a function f ∈ QC ρZ -w ∗ ∩ Σ Λ such that Gr ρZ ×w∗f = GrΓ.(iii) Σ Λ ⊂ QC ρZ -w ∗and for all f ∈ Σ Λ, Gr ρZ ×w∗f = GrΓ.45