Graph topologies and uniform convergence in quasi-uniform spaces. (English) Zbl 0995.54007

The authors generalize some results from their article [Houston J. Math. 27, No. 2, 445-459 (2001; Zbl 0992.54017)] to quasi-uniform spaces. Among other things they prove that if \((X,{\mathcal U})\) is a quasi-uniform space and \((L,{\mathcal V})\) a Scott quasi-uniform semigroup, then the following statements are equivalent:
(i) Every continuous function from \(X\) to \(L\) is quasi-uniformly continuous.
(ii) The proximal topology of \((X\times L\), \({\mathcal U}^{-1}\times {\mathcal V})\) agrees with the topology of uniform convergence on the set \(C(X,L)\) of continuous functions from \(X\) to \(L.\)
(iii) The upper Hausdorff quasi-uniform topology induced by \({\mathcal U}^{-1}\times {\mathcal V}\) agrees with the topology of uniform convergence on \(C(X,L).\)
Here a Scott quasi-uniform semigroup is a pair \((L,{\mathcal U})\) where \(L\) is a complete lattice, \({\mathcal U}\) is a quasi-uniformity on \(L\) such that \({\mathcal T}({\mathcal U})\) is the Scott topology and \((L,\vee,{\mathcal T}({\mathcal U}))\) is a semitopological semigroup, that is, the left translations are continuous. Furthermore, continuous functions are identified with their graphs.


54B20 Hyperspaces in general topology
54E15 Uniform structures and generalizations
06B30 Topological lattices


Zbl 0992.54017