Galvin, Fred; Shore, S. D. Completeness in semimetric spaces. (English) Zbl 0558.54019 Pac. J. Math. 113, 67-75 (1984). 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 Cited in 17 Documents MSC: 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 PDFBibTeX XMLCite \textit{F. Galvin} and \textit{S. D. Shore}, Pac. J. Math. 113, 67--75 (1984; Zbl 0558.54019) Full Text: DOI