Short paths in homogeneous continua
A homogeneous continuum can be almost arcwise connected without being locally connected, e.g., the dyadic solenoid. There are many known examples of arcwise connected homogeneous continua and each of them is locally connected. K. Kuperberg has asked whether every such continuum must be locally connected. If so, then every such continuum must have all of the properties of locally connected continua. (Of course, being arcwise connected is one such property.) In this paper the author discovers another. If X is a homogeneous arcwise connected continuum and \(\epsilon\) is a positive number, then there exists a natural number n such that any two points of X can be joined in X by an \(\epsilon\)-chain of n subarcs. The author then raises the question: Does this imply that there exists a family P of paths in X such that (1) for x,y\(\in X\) some path in P joins x to y and (2) for every \(\epsilon >0\) there exists a natural number n such that each path in P is the union of the element of an \(\epsilon\)- chain of n elements? If so, is homogeneity necessary?
scientific article; zbMATH DE number 4005134