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

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 .