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Completeness in semimetric spaces. (English) Zbl 0558.54019
This interesting paper compares various forms of completeness in semimetric spaces in face of certain ”continuity properties” of distance functions. Two such properties are developability: lim d(x${}_{n},p\right)=limd\left({y}_{n},p\right)=0$ implies lim d(x${}_{n},{y}_{n}\right)=0$, and 1- continuity: for any q, lim d(x${}_{n},p\right)=0$ implies lim d(x${}_{n},q\right)=d\left(p,q\right)$. And two of the authors’ main results are as follows. Theorem: For any 1-continuous semimetric d, a semimetrizable space is d- Cauchy complete if and only if it is d-weakly complete in the sense of L. F. McAuley (ibid. 6, 315-326 (1956; Zbl 0072.178)]. Theorem: A semimetrizable space may be Cauchy complete and developable and yet admit no semimetric which is (simultaneously) Cauchy complete and developable.
Reviewer: P.J.Collins
##### MSC:
 5.4e+26 Semimetric spaces