## 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$${}_ n,p)=\lim d(y_ n,p)=0$$ implies lim d(x$${}_ n,y_ n)=0$$, and 1- continuity: for any q, lim d(x$${}_ n,p)=0$$ implies lim d(x$${}_ n,q)=d(p,q)$$. 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 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

Zbl 0072.178
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