Completeness in semimetric spaces (Q762794)

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 \textit{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.