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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extending a complete convex metric


Author: Robert A. Dooley
Journal: Proc. Amer. Math. Soc. 34 (1972), 553-559
MSC: Primary 54E50; Secondary 52A50
MathSciNet review: 0298627
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0298627-6
PII: S 0002-9939(1972)0298627-6
Keywords: Convex metric, generalized continuum, strongly convex, without ramifications, simple triod, interval, simple closed curve
Article copyright: © Copyright 1972 American Mathematical Society