9.5 Dense SelectionsFor each F ∈ P ρZ ×w ∗ f(Z × X) with induced USCOthere is a single-valued set given byz → F (z) :={x ∈ X :(z,x) ∈ F } , (79)S F := {z ∈ Z : F (z) is a singleton} . (80)We will denote by DSF f (Z × X) the collection of all densely single-valued forms,that is, the collection of all sets F ∈ P ρZ ×w ∗ f(Z × X) such that S F is ρ Z -dense in Z.Thus, the collection DSF f := DSF f (Z × X) is given byDSF f (Z × X) := © F ∈ P ρZ ×w ∗ f(Z × X) :S F is ρ Z -dense in Z ª . (81)It follows from Proposition 1 in Crannell, Frantz, <strong>and</strong> LeMasurier (2005) that if a setF ∈ P ρZ ×w ∗ f(Z × X) is such that F = Gr ρZ ×w∗f for some function f : Z → X, thenS F = C ρZ -w ∗(f) =S Gr ∗f. Thus, the single-valued set, S ρZ ×w Gr ρZ ×w∗f, correspondingto any function f : Z → X, or the single-valued set, S F , corresponding to any setF ∈ P ρZ ×w ∗ f(Z ×X) such that F = Gr ρZ ×w ∗f is equal to the set of ρ Z-w ∗ -continuitypoints, C ρZ -w∗(f), of the function f generating that set via graph closure. Moreover,given F ∈ P ρZ ×w ∗ f(Z × X), iff(·) is a selection of F (·), the closure of whose graph isstrictly contained in F ,thenS F ⊂ C ρZ -w ∗(f), <strong>and</strong>ifg is a selection of Gr ρ Z ×w ∗f(·),then C ρZ -w ∗(f) ⊂ C ρ Z -w ∗(g).Let F ∈ P f (Z × X) with induced USCO F (·) ∈ U ρZ -w ∗ := U(Z, P w ∗ f(X)). Wesay that a function, f : Z → X, is a selection of F (·) if f(z) ∈ F (z) for all z ∈ Z.Definition 10 (Dense Selections)We say that f is a dense selection of USCO F (·) ∈ U ρZ -w ∗F (·) <strong>and</strong>Gr ρZ ×w∗f = GrF.if f is a selection ofWe close our brief discussion of USCO mappings with a result by Beer (1983)characterizing dense selections. Recall that in a topological space a point z is isolatedif {z}∩U z = {z} for all neighborhoods U z of z. Thepointz is a limit point if for eachneighborhood U z of z contains a point z 0 (6= z). The following result, characterizingUSCOs with dense selections, is an immediate consequence of Theorem 1 in Beer(1983). In our statement of Beer’s result we take as given the fact that Z <strong>and</strong> Xare compact metric spaces, equipped with convex metrics, ρ Z on Z compatible withthe metric d Z <strong>and</strong> ρ w ∗ on X compatible with the weak star topology in X. Infact,Beer’s result requires only that Z be a complete separable metric space (i.e., a Polishspace) <strong>and</strong> that X be a sigma compact complete, separable metric space.48
Lemma 22 (A characterization of USCO correspondences with dense selections,Beer, 1983)Suppose [A-1] holds. Let Γ ∈ U ρZ -w ∗ := U(Z, P w ∗ f(X)). The following statementsare equivalent.(a) Γ has a dense selection.(b) Γ has the following properties:(b-1) For each z ∈ Z the set {(z, Γ(z))} := {(z,x) :x ∈ Γ(z)} includes at most oneisolated point of GrΓ;(b-2) For each (z, x) ∈ {(z,Γ(z))}, (z,x) is not a limit point of GrΓ\{(z, Γ(z))} if<strong>and</strong> only if (z, x) is an isolated point of GrΓ.Let X = Y =[0, 2] <strong>and</strong> consider the USCO, Λ ∈ U := U(X, P f (Y )), givenbyΛ(x) =⎧⎨⎩{0} 0 ≤ x
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