A convex metric for a locally connected continuum

Abstract

A subset of a topological space 5 is said to have a convex metric (even though S may have no metric) if the subspace of 5 has a convex metric. It is known [5 J that a compact continuum is locally connected if it has a convex metric. The question has been raised [5] as to whether or not a compact locally connected continuum can be assigned a convex metric. Menger showed [5] that is convexifiable if it possesses a metric such that for each point p of and each positive number e there is an open subset R of containing p such that each point of R can be joined in to p by a rectifiable arc of length (under D) less than e. Kuratowski and Whyburn proved [4] that has a convex metric if each of its cyclic elements does. Beer considered [ l ] the case where is one-dimensional. Harrold found [3] to be convexifiable if it has the additional property of being a plane continuum with only a finite number of complementary domains. We shall show that if M and M2 are two intersecting compact continua with convex metrics D and D2 respectively, then there is a convex metric D% on M\-\-Mi that preserves D on M (Theorem 1). Using this result, we show that any compact ^-dimensional locally connected continuum has a convex metric (Theorem 6). We do not