Externalities, Nonconvexities, and Fixed Points

denote the collection of all densely continuous forms. Thus, DCF f (Z × X) is givenbyDCF f (Z × X)o:=nE ∈ P ρZ ×w ∗ f(Z × X) :E = Gr ρZ ×w ∗f| C ρZ -w ∗ (f) for some f ∈ DC ρZ -w ∗ .(78)The following two part Lemma summarizes what we need to know about denselycontinuous forms. Part (1) of the Lemma, due to Crannell, Franz, and LeMasurier(2005), tells us that all selections from a densely continuous form share the same setof continuity points, and moreover, that all such selections are equal on this set ofcontinuity points. Part (2) is due to Hola and Holy (2009) and tells us that for anyselection from a densely continuous form, the graph closure of the selection restrictedto its set of continuity points is equal to the graph closure of the selection itself.Thus, together Parts (1) and (2) tell us that the graph closures of all selections froma densely continuous form are equal.In our statement of these results we take as given the fact that Z and X arecompact metric spaces, equipped with convex metrics, ρ Z and ρ w ∗, compatible withthe d Z topology in Z and with the weak star topology in X. In fact, Part (1) due toCrannell, Franz, and LeMasurier (2005) requires that Z and X be compact metricspaces. Part (2) continues to hold if Z and X are not compact - but requires that Zbe a metric Baire space and X a metric space.Lemma 21 (Minimal USCOs and Densely Continuous Forms)(1) (Crannell, Franz, and LeMasurier, 2005)If GrΛ ∈ DCF f (Z × X) and f ∈ Σ Λ ∩ QC ρZ -w ∗, then for all g ∈ Σ Λ∩ QC ρZ -w ∗,C ρZ -w ∗(f) =C ρ Z -w ∗(g) and Grf| C ρZ -w ∗(f) = Grg| CρZ -w ∗ (g) .(1) (Hola and Holy, 2009)The following statements are equivalent.(i) Λ ∈ M ρZ -w∗(Z, X).(ii) GrΛ ∈ DCF f (Z × X).(iii) If f ∈ Σ Λ ,thenGrf| CρZ -w ∗(f) = Grf.(iv) If f ∈ Σ Λ , then for every z ∈ Z and every neighborhood U of (z, f(z)) thereexists z 0 ∈ C ρZ -w ∗(f) such that (z0 ,f(z 0 )) ∈ U.47