Extending a complete convex metric

Dooley, Robert A.

doi:10.1090/S0002-9939-1972-0298627-6

Extending a complete convex metric
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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.

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