Ultrametrically injective spaces

Bayod, José M.; Martínez-Maurica, J.

doi:10.1090/S0002-9939-1987-0908671-1

Ultrametrically injective spaces

Authors: José M. Bayod and J. Martínez-Maurica
Journal: Proc. Amer. Math. Soc. 101 (1987), 571-576
MSC: Primary 54E35; Secondary 54E40
DOI: https://doi.org/10.1090/S0002-9939-1987-0908671-1
MathSciNet review: 908671
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the injective objects in the category of ultrametric spaces with contractive mappings. As a result, the property of spherical completeness is characterized in several new ways related to extensions of uniformly continuous mappings and to extensions of isometries. A construction of the injective envelope is also given.

[1] N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439. MR 0084762
[2] José M. Bayod, A spherical completion within the infinitesimal hull of an ultrametric space, Quaestiones Math. 7 (1984), no. 3, 241–249. MR 771651
[3] Andreas W. M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. in Math. 53 (1984), no. 3, 321–402. MR 753872, https://doi.org/10.1016/0001-8708(84)90029-X
[4] Robert L. Ellis, Extending uniformly continuous pseudo-ultrametrics and uniform retracts, Proc. Amer. Math. Soc. 30 (1971), 599–602. MR 0283752, https://doi.org/10.1090/S0002-9939-1971-0283752-5
[5] J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76. MR 0182949, https://doi.org/10.1007/BF02566944
[6] John R. Isbell, 𝑠 admits an injective metric, Proc. Amer. Math. Soc. 28 (1971), 259–261. MR 0282333, https://doi.org/10.1090/S0002-9939-1971-0282333-7
[7] Jie Hua Mai and Yun Tang, An injective metrization for collapsible polyhedra, Proc. Amer. Math. Soc. 88 (1983), no. 2, 333–337. MR 695270, https://doi.org/10.1090/S0002-9939-1983-0695270-9
[8] Grattan P. Murphy, A metric basis characterization of Euclidean space, Pacific J. Math. 60 (1975), no. 2, 159–163. MR 0394446
[9] R. Rammal, G. Toulouse, and M. A. Virasoro, Ultrametricity for physicists, Rev. Modern Phys. 58 (1986), no. 3, 765–788. MR 854445, https://doi.org/10.1103/RevModPhys.58.765
[10] W. H. Schikhof, Isometrical embeddings of ultrametric spaces into non-Archimedean valued fields, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 1, 51–53. MR 748978
[11] J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Springer-Verlag, New York-Heidelberg, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. MR 0461107