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Zbl 0558.54019
Galvin, Fred; Shore, S.D.
Completeness in semimetric spaces.
(English)
[J] Pac. J. Math. 113, 67-75 (1984). ISSN 0030-8730

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${}\sb n,p)=\lim d(y\sb n,p)=0$ implies lim d(x${}\sb n,y\sb n)=0$, and 1- continuity: for any q, lim d(x${}\sb n,p)=0$ implies lim d(x${}\sb n,q)=d(p,q)$. 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 {\it 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.
[P.J.Collins]
MSC 2000:
*54E25 Semimetric spaces

Keywords: Cauchy completeness; strong and weak completeness; Moore completeness in semimetrizable spaces; developable semimetrizable space; 1-continuously semimetrizable space

Citations: Zbl 0072.178

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