There is a continuum which is connected by uniformly short paths but not uniformly path connected

Abstract

David Bellamy defined the concept of connectedness by uniformly short paths and asked wether it was equivalent to the notion of uniform path connectedness. A counterexample is described. Some definitions: for a positive real number e, a Peano continuum has “e-length” ≤ an integer n iff it is the union of a chain of n Peano continua of diameter less than e. A continuum (compact connected metric space) X is “uniformly path connected” iff there is a family P of paths such that for each a, b in X, there is a path in P from a to b, and for each e there is an N such that the e-length of the image of each path in P is ≤ N. A continuum is “connected by uniformly short paths” iff for each e there is a number N such that for each pair of points a, b in X there is a path from a to b whose image has e-length ≤ N. (The definition of e-length given is more concise than that given in the paper).