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Further extending a complete convex metric


Author: Robert A. Dooley
Journal: Proc. Amer. Math. Soc. 40 (1973), 590-596
MSC: Primary 54E50
DOI: https://doi.org/10.1090/S0002-9939-1973-0322824-5
MathSciNet review: 0322824
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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, metric space. Let $ {M_1}$ be a nonempty space with a complete convex metric $ {D_1}$ and let $ {M_2}$ be a nonempty locally connected generalized continuum. The following condition is shown to be necessary and sufficient for there to exist a complete convex metric for $ {M_1} \cup {M_2}$ that extends $ {D_1}:{M_1} \cap {M_2}$ is a nonempty subspace of both $ {M_1}$ and $ {M_2}$ which is closed in $ {M_2}$ and whose $ {M_2}$ boundary is closed in $ {M_1}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0322824-5
Keywords: Convex metric, generalized continuum
Article copyright: © Copyright 1973 American Mathematical Society

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