Graph Topologies and Uniform Convergence in Quasi-Uniform ...

graph topologies and quasi-uniform convergence 133Every quasi-uniformity U on X generates a topology T (U). A neighborhoodbase for each point x ∈ X is given by {U(x) : U ∈ U} whereU(x) = {y ∈ X : (x, y) ∈ U}.Similarly, the topology generated on a set X by a quasi-pseudo-metric d,is denoted by T (d).In the sequel, we shall denote by l the quasi-pseudo-metric on R definedby l(x, y) = (x − y) ∨ 0 for all x, y ∈ R.For any nonempty set X, we denote by P 0 (X) the family of all nonemptysubsets of X.2. PreliminariesGiven two quasi-uniform spaces (X, U) and (Y, V), a function f from Xto Y is said to be quasi-uniformly continuous if for every V ∈ V we can findU ∈ U such that if (x, y) ∈ U then (f(x), f(y)) ∈ V .The set of all quasi-uniformly continuous functions from X to Y is denotedby UC(X, Y ).Definition 1. Let (X, U) be a quasi-uniform space.[11], we defineFollowing [5] andU + H = {(A, B) ∈ P 0(X) × P 0 (X) : B ⊆ U(A)}U − H = {(A, B) ∈ P 0(X) × P 0 (X) : A ⊆ U −1 (B)}for all U ∈ U. Then {U + H: U ∈ U} is a base for the upper Hausdorff quasiuniformityH + U of U and {U − H: U ∈ U} is a base for the lower Hausdorffquasi-uniformity H − U of U. The quasi-uniformity H U defined as the supremumof the lower and upper Hausdorff quasi-uniformities is called the Hausdorffquasi-uniformity of U.Given two quasi-uniform spaces (X, U) and (Y, V) we denote by C(X, Y )the family of continuous functions from (X, T (U)) to (Y, T (V)). If f ∈C(X, Y ) we denote the graph of f by Gr f, i.e. Gr f = {(x, f(x)) : x ∈ X}.On the set C(X, Y ) we consider the upper Hausdorff quasi-uniformityH + U −1 ×Vinduced by the product quasi-uniformity U −1 × V when we identifyeach continuous function with its graph. Then, each basic entourageof H + U −1 ×V is of the form (U −1 × V ) + H= {(f, g) ∈ C(X, Y ) × C(X, Y ) :(U × V −1 )(x, g(x)) ∩ Gr f ≠ ∅ for all x ∈ X}, where U ∈ U and V ∈ V.

134 j. rodríguez-lópez, s. romagueraSimilarly, the lower Hausdorff quasi-uniformity H − on C(X, Y ) hasU −1 ×Vbasic entourages of the form (U −1 × V ) − H= {(f, g) ∈ C(X, Y ) × C(X, Y ) :(U −1 × V )(x, f(x)) ∩ Gr g ≠ ∅ for all x ∈ X}, where U ∈ U and V ∈ V.The Hausdorff quasi-uniformity H U −1 ×V induced on C(X, Y ) by the productquasi-uniformity U −1 × V is the supremum of H + U −1 ×V and H− U −1 ×V .Definition 2. Let (X, U) and (Y, V) be two quasi-uniform spaces. Thetopology generated on C(X, Y ) by the sets of the form G − = {f ∈ C(X, Y ) :Gr f ∩ G ≠ ∅} where G is a T (U −1 × V)-open set is called the lower proximaltopology and is denoted by T − (δ U −1 ×V).The topology generated on C(X, Y ) by all sets of the form G ++ = {f ∈C(X, Y ) : there exists U ∈ U and V ∈ V such that (U −1 × V )(Gr f) ⊆ G}where G is a T (U −1 × V)-open set is called the upper proximal topology andis denoted by T + (δ U −1 ×V).The topology T (δ U −1 ×V) = T + (δ U −1 ×V) ∨ T − (δ U −1 ×V) is called the proximaltopology.Let us recall that the proximal topology was essentially introduced in [13],although the term “proximal (hit-and-miss) topology” was firstly used in [4],[6] and [12], where the relationship between the proximal topology and otherknown hyperspace topologies was studied. In particular, the proximal topologyis compatible with Fisher convergence [7] of sequences of sets, as consideredin [1].3. The resultsProposition 1. Let (X, U) and (Y, V) be two quasi-uniform spaces. ThenT (δ U −1 ×V) = T (H + ) on C(X, Y ).U −1 ×VProof. Let us consider the set G ++ where G is a T (U −1 × V)-open setand A ∈ G ++ . Therefore, we can find U ∈ U and V ∈ V such that (U −1 ×V )(A) ⊆ G. Let us consider the set (U −1 × V ) + H(A). It is evident thatA ∈ (U −1 × V ) + H (A) ⊆ G++ .Furthermore, if we have that Gr f ∈ G − where G is a T (U −1 × V)-openset and f ∈ C(X, Y ), let (x, f(x)) ∈ Gr f ∩ G. Since (x, f(x)) ∈ G we canfind U ∈ U and V ∈ V verifying (U −1 × V )((x, f(x))) ⊆ G. Let W ∈ V suchthat W 2 ⊆ V . Since f ∈ C(X, Y ) we can find U ′ ∈ U such that (f(x), f(y)) ∈W whenever (x, y) ∈ U ′ . We claim that (U ′−1 × W ) + H (Gr f) ⊆ G− . LetGr g ∈ (U ′−1 × W ) + H(Gr f). Therefore, we can find (y, f(y)) ∈ Gr f such that

graph topologies and quasi-uniform convergence 135(x, y) ∈ U ′ and (f(y), g(x)) ∈ W . Furthermore, by the above observationwe obtain that (f(x), f(y)) ∈ W , so (f(x), g(x)) ∈ W 2 ⊆ V . Consequently,(x, g(x)) ∈ (U −1 × V )(x, f(x)) ⊆ G, i.e. Gr g ∩ G ≠ ∅.For the other inclusion, let us consider the set (U −1 × V ) + H(A) whereU ∈ U and V ∈ V. It is easy to prove that (int T (U −1 ×V)((U −1 × V )(A))) ++ ⊆(U −1 × V ) + H (A).If (X, U) and (Y, V) are two quasi-uniform spaces, the quasi-uniformity ofuniform convergence on C(X, Y ) induced by V is the quasi-uniformity V X onC(X, Y ) whose elements V X are defined byV X = {(f, g) ∈ C(X, Y ) × C(X, Y ) : (f(x), g(x)) ∈ V for all x ∈ X}.The topology generated by V X is called, simply, the topology of uniformconvergence on C(X, Y ) and is denoted by T (V X ).Proposition 2. Let (X, U) and (Y, V) be two quasi-uniform spaces. ThenT (H U −1 ×V) ⊆ T (V X ) on C(X, Y ).Proof. Let us suppose that {f λ } λ∈Λ is a net in C(X, Y ) which is T (V X )-convergent to f ∈ C(X, Y ). Consider the set (U −1 × V ) H (Gr f) where U ∈ Uand V ∈ V. Therefore, we can find λ 0 ∈ Λ such that f λ ∈ V X (f) for allλ ≥ λ 0 . It is evident that Gr f λ ∈ (U −1 × V ) H (Gr f) for all λ ≥ λ 0 .The following proposition extends Naimpally’s theorem to the quasiuniformsetting.Proposition 3. Let (X, U) and (Y, V) be two quasi-uniform spaces. ThenT (δ U −1 ×V) = T (H + U −1 ×V ) = T (H U −1 ×V) = T (V X ) on UC(X, Y ).Proof. By the above results, we only have to show that T (V X ) ⊆T (H + U −1 ×V ).Let {f λ } λ∈Λ be a net in UC(X, Y ) such that it is T (H + U −1 ×V )-convergentto a function f ∈ UC(X, Y ). Fix W ∈ V. Therefore, we can find U ∈ Usuch that if (x, y) ∈ U then (f(x), f(y)) ∈ W ′ where W ′ ∈ V and W ′2 ⊆ W .Furthermore, there exists λ 0 ∈ Λ such that Gr f λ ∈ (U −1 × W ′ ) + H(Gr f) forall λ ≥ λ 0 .Given x ∈ X and λ ≥ λ 0 we can find y ∈ X such that (x, y) ∈ U and(f(y), f λ (x)) ∈ W ′ . Therefore, (f(x), f λ (x)) ∈ W ′2 ⊆ W , i.e. f ∈ W X (f λ ) forall λ ≥ λ 0 .

136 j. rodríguez-lópez, s. romagueraNow, we present some examples where we can apply this result.Example 1. As we observed at the introduction, if f : (X, T ) → (Y, T ′ )is a continuous function and U denotes the Pervin (resp. point finite, locallyfinite, semi-continuous, fine transitive, fine) quasi-uniformity on X (see [8] forthe corresponding definitions) and V the corresponding quasi-uniformity onY then f : (X, U) → (Y, V) is quasi-uniformly continuous (see [8, Proposition2.17]). Therefore, we deduce that T (H U −1 ×V) = T (V X ) on C(X, Y ).Now, we recall some definitions about the theory of lattices. Our mainreference is [9].A partially ordered set (or a poset) is a nonempty set L with a reflexive,transitive and antisymmetric relation ≤. A lattice is a poset where everynonempty finite subset has infimum and supremum.Given a lattice L and O ⊆ L, we denote ↑ O = {l ∈ L : there exists o ∈O such that o ≤ l}.A lattice is said to be complete if every nonempty subset has infimum andsupremum.Let L be a complete lattice. We say that x is way below y, and we denoteit by x ≪ y, if for all directed subsets D of L the relation y ≤ sup D impliesthe existence of d ∈ D such that x ≤ d, where D is called directed set if givenp, q ∈ D we can find l ∈ D such that p ≤ l and q ≤ l.A continuous lattice is a complete lattice L such that it satisfies the axiomof approximation:x = sup{u ∈ L : u ≪ x}for all x ∈ L.Let L be a complete lattice. The Scott topology σ(L) is formed by allsubsets O of L which satisfy:i) O =↑ Oii) sup D ∈ O implies D ∩ O ≠ ∅ for all directed sets D ⊆ L.We recall that a semigroup is a pair (X, ·) such that · is an associativeinternal law or operation on X for which exists an identity element e.Definition 3. If (X, T ) is a topological space and (X, ·) is a semigroupsuch that the function ·x : X → X defined by ·x(y) = x · y for all y ∈ X iscontinuous for all x ∈ X we say that (X, ·, T ) is a semitopological semigroup.

graph topologies and quasi-uniform convergence 137Definition 4. A Scott quasi-uniform semigroup is a pair (L, U) where Lis a complete lattice, U is a quasi-uniformity on L such that T (U) is the Scotttopology and (L, ∨, T (U)) is a semitopological semigroup.Now, we prove our main result.Theorem 1. Let (X, U) be a quasi-uniform space and (L, V) a Scottquasi-uniform semigroup. The following statements are equivalent:i) Every continuous function from X to L is quasi-uniformly continuous.ii) The proximal topology of (X × L, U −1 × V) agrees with the topology ofuniform convergence on C(X, L).iii) The upper Hausdorff quasi-uniform topology induced by U −1 × V agreeswith the topology of uniform convergence on C(X, L).Proof. i) ⇒ ii) This is deduced from Proposition 3.ii) ⇒ iii) We only have to use Proposition 1.iii) ⇒ i) Suppose that there is a continuous function f : (X, T (U)) →(L, σ(L)) which is not quasi-uniformly continuous. Then, we can findW ∈ V and two nets {a U } U∈U and {b U } U∈U such that (a U , b U ) ∈ U and(f(a U ), f(b U )) ∉ W for all U ∈ U. For each U ∈ U we define the followingfunction f U : X → L:⎧⎨f(b U ) if x ∈ {a U }f U (x) =⎩f(x) ∨ f(b U ) otherwise.Let us prove that f U is continuous for each U ∈ U. Let us fix U ∈ U andlet x ∈ X and {x λ } λ∈Λ be a T (U)-convergent net to x. We distinguish thefollowing cases:Case 1. x ∈ {a U }We have that f U (x) = f(b U ), and the values of f U (x λ ) can be f(b U ) orf(x λ ) ∨ f(b U ) for all λ ∈ Λ. It is evident that in both cases we obtain thatf U (x) ≤ f U (x λ ), so f U is a continuous function in x.Case 2. x ∈ X\{a U }Since {x λ } λ∈Λ is T (U)-convergent to x, it is eventually in X\{a U }. Therefore,by continuity of the sup operation, we deduce that f U is continuous in x.Consequently, we have shown that f U is continuous for all U ∈ U.We now prove that {f U } U∈U converges to f with respect to T (H + U −1 ×V ).

138 j. rodríguez-lópez, s. romagueraFix U 0 ∈ U and let U, V ∈ U such that V 2 ⊆ U ⊆ U 0 . If x ∈ {a V },f V (x) = f(b V ). Furthermore, (x, a V ) ∈ V , so (x, b V ) ∈ V 2 ⊆ U.On the other hand, if x ∈ X\{a V } then f(x) ≤ f V (x) = f(x) ∨ f(b V ).Hence {f U } U∈U converges to f with respect to T (H + ). However, thisU −1 ×Vnet does not converge to f with respect to the topology of uniform convergencebecause (f(a U ), f U (a U )) = (f(a U ), f(b U )) ∉ W for all U ∈ U.Remark 1. As a particular case of the above theorem we can consider thecontinuous lattice R ∗ with the usual order where R ∗ = R ∪ {−∞, +∞}. Wecan consider the Scott topology σ(R ∗ ) of this continuous lattice and a quasiuniformityU on R ∗ such that T (U) = σ(R ∗ ). In every continuous lattice thesup operation is continuous considering the Scott topology (see [9, Chapter I,Proposition 1.11]). Therefore, we can apply the above theorem to this lattice.In particular, we can consider the extended lower quasi-pseudo-metric onR ∗ given by⎧⎪⎨ (x − y) ∨ 0 if x, y ∈ Rl ∗ (x, y) = 0 if x = −∞ or y = +∞⎪⎩+∞ if x = +∞ or y = −∞ and x ≠ y.The topology generated by this extended quasi-pseudo-metric has as abase all the sets of the form (a, +∞] = {b ∈ R ∗ : a ≪ b} where a ∈ R ∗and the open set {+∞} which coincide with the basic open sets in the Scotttopology (see [9, Chapter I, Proposition 1.10]). Now, the continuous functionsbetween X and R ∗ are the extended lower semicontinuous functions, i.e. lowersemicontinuous functions with values in R ∗ (see [3, 9]). Therefore, the aboveresult asserts that given a quasi-uniform space (X, U) then every extendedlower semicontinuous function on X is quasi-uniformly continuous if and onlyif the topology of uniform convergence agrees with the upper Hausdorff quasiuniformtopology induced by U −1 ×U l ∗ where U l ∗ denotes the quasi-uniformityinduced by l ∗ .Remark 2. We notice that the above theorem is also true if we considerfunctions from a quasi-uniform space to a lattice contained in a Scott quasiuniformsemigroup. Therefore, we not only can consider the extended realline R ∗ but the real line R. Consequently, given a quasi-uniform space (X, U)then SC(X) = UC(X, R) if and only if the topology of uniform convergencecoincides with the upper Hausdorff quasi-uniform topology induced byU −1 × U l .

graph topologies and quasi-uniform convergence 139We recall that a T 0 -space Z is called injective if every continuous mapf : X → Z extends continuously to any space Y containing X as a subspace.Corollary 1. Let (X, U) be a quasi-uniform space and (Y, τ) an injectivetopological T 0 -space. Let V be a quasi-uniformity compatible with τ. Thefollowing statements are equivalent:byi) Every continuous function from X to Y is quasi-uniformly continuous.ii) The proximal topology of (X × Y, U −1 × V) agrees with the topology ofuniform convergence on C(X, Y ).iii) The upper Hausdorff quasi-uniform topology induced by U −1 × V agreeswith the topology of uniform convergence on C(X, Y ).Proof. If Y is a T 0 space it is evident that the specialization order ≤ givenx ≤ y ⇔ x ∈ {y}is a partial order and it can be proved (see [9, Chapter II, Theorem 3.8]) thatif Y is an injective space then Y with the specialization order is a continuouslattice. Furthermore, in a continuous lattice the sup operation is jointlycontinuous with respect to the Scott topology (see [9, Chapter I, Proposition1.11]), so we can apply the theorem above, but the Scott topology (see [9,Chapter II, Theorem 3.8]) is homeomorphic to τ so we have completed theproof.AcknowledgementsThe authors are grateful to the referee, in particular for suggesting ashorter proof of Theorem 1.References[1] Baronti, M., Papini, P., Convergence of sequences of sets, in “Methodsof Functional Analysis in Approximation Theory”, ISNM 76, Birkhäuser-Verlag, 1986, 135 – 155.[2] Beer, G., Metric spaces on which continuous functions are uniformly continuousand Hausdorff distance, Proc. Amer. Math. Soc., 95 (1985), 653 – 658.[3] , “Topologies on Closed and Closed Convex Sets”, vol. 268, KluwerAcademic Publishers, Dordrecht, 1993.[4] Beer, G., Lechicki, A., Levi, S., Naimpally, S., Distance functionalsand suprema of hyperspace topologies, Ann. Mat. Pura Appl., 162 (1992),367 – 381.[5] Berthiaume, G., On quasi-uniformities in hyperspaces, Proc. Amer. Math.Soc., 66 (1977), 335 – 343.

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