Extending a complete convex metric
HTML articles powered by AMS MathViewer
- by Robert A. Dooley PDF
- Proc. Amer. Math. Soc. 34 (1972), 553-559 Request permission
Abstract:
A metric D is convex if for every two points x, z there is a third point y such that $D(x,y) + D(y,z) = D(x,z)$. A generalized continuum is a connected, locally compact, separable metric space. It is shown that if ${M_1}$ is a space with a complete convex metric ${D_1}$ and ${M_2}$ is a locally connected generalized continuum whose intersection with ${M_1}$ is nonempty and compact, there is a complete convex metric for ${M_1} \cup {M_2}$ that extends ${D_1}$. Using this result, four classes of locally connected generalized continua are characterized by the type of complete convex metric they admit.References
- Anthony D. Berard Jr., Characterizations of metric spaces by the use of their midsets: Intervals, Fund. Math. 73 (1971/72), no. 1, 1–7. MR 295300, DOI 10.4064/fm-73-1-1-7
- R. H. Bing, A convex metric for a locally connected continuum, Bull. Amer. Math. Soc. 55 (1949), 812–819. MR 31712, DOI 10.1090/S0002-9904-1949-09298-4
- R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc. 55 (1949), 1101–1110. MR 35429, DOI 10.1090/S0002-9904-1949-09334-5
- Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
- Karol Borsuk, On a metrization of polytopes, Fund. Math. 47 (1959), 325–341. MR 112122, DOI 10.4064/fm-47-3-325-341
- A. Lelek and Jan Mycielski, On convex metric spaces, Fund. Math. 61 (1967), 171–176. MR 221468, DOI 10.4064/fm-61-2-171-176
- A. Lelek and W. Nitka, On convex metric spaces. I, Fund. Math. 49 (1960/61), 183–204. MR 124882, DOI 10.4064/fm-49-2-183-204
- Karl Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 100 (1928), no. 1, 75–163 (German). MR 1512479, DOI 10.1007/BF01448840
- Edwin E. Moise, Grille decomposition and convexification theorems for compact metric locally connected continua, Bull. Amer. Math. Soc. 55 (1949), 1111–1121. MR 35430, DOI 10.1090/S0002-9904-1949-09336-9
- Akira Tominaga and Tadashi Tanaka, Convexification of locally connected generalized continua, J. Sci. Hiroshima Univ. Ser. A 19 (1955), 301–306. MR 78677
- Fausto A. Toranzos, Spaces that admit only a certain type of convex metric, Nieuw Arch. Wisk. (3) 14 (1966), 252–254. MR 205225
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 34 (1972), 553-559
- MSC: Primary 54E50; Secondary 52A50
- DOI: https://doi.org/10.1090/S0002-9939-1972-0298627-6
- MathSciNet review: 0298627