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Left \(K\)-completeness in quasi-metric spaces. (English) Zbl 0784.54027
A sequence \(\langle x_ n\rangle\) in a quasi-pseudo-metric space \((X,d)\) is weakly left K-Cauchy provided that for each \(\varepsilon>0\) there exists \(k\in\mathbb{N}\) such that \(d(x_ k,x_ n)<\varepsilon\) whenever \(k\leq n\). A quasi-pseudo-metric space \((X,d)\) is weakly left K- sequentially complete provided that each weakly left K-Cauchy sequence in \(X\) converges in \({\mathcal T}(d)\). The author characterizes regular left K- sequentially complete quasi-metric spaces by means of certain bases of countable order. It follows that these spaces are complete Aronszajn spaces.

MSC:
54E15 Uniform structures and generalizations
54E50 Complete metric spaces
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