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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.

MSC:

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

Citations:

Zbl 0992.54017
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