A Convex Metric for a Locally Connected Continuum

A subset M of a topological space 5 is said to have a convex metric (even though S may have no metric) if the subspace M 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 M can be assigned a convex metric. Menger showed [5] that M is convexifiable if it possesses a metric D such that for each point p of M and each positive number e there is an open subset R of M containing p such that each point of R can be joined in M to p by a rectifiable arc of length (under D) less than e. Kuratowski and Whyburn proved [4] that M has a convex metric if each of its cyclic elements does. Beer considered [ l ] the case where M is one-dimensional. Harrold found [3] M 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