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The coregular property on \(\gamma \)-spaces. (English) Zbl 0949.54031
A \(\gamma \)-space stands for a T\(_1\) locally quasi-uniform space with a countable base. A \(\gamma \)-space conjecture says that every \(\gamma \)-space is quasi-metrizable. Since Fox’s counterexample from 1982, disproving this conjecture, many authors worked on a task to characterize those \(\gamma \)-spaces which are quasi-metrizable. The paper under review focuses on providing some sufficient conditions on a \(\gamma \)-space to be quasi-metrizable. We select two from the results contained in the paper.
Theorem. If \((X,{\mathcal U})\) is a \(\gamma \)-space, the space \((X,{\mathcal U}^{-1},{\mathcal U})\) is coregular, and the space \((X,{\mathcal U}^*)\) is Lindelöf, then the space \((X,{\mathcal U})\) is quasi-metrizable.
Theorem. If \((X,{\mathcal U})\) is a \(\gamma \)-space such that both \(\mathcal U\) and \({\mathcal U}^{-1}\) are generalized quasi-uniformities, and \(\tau ({\mathcal U}^{-1}) \subset \tau ({\mathcal U})\), then the space \((X,{\mathcal U})\) is quasi-metrizable.
Reviewer: M.Fabián (Praha)

MSC:
54E15 Uniform structures and generalizations
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