Widely-connected sets in the bucket-handle continuum
Volume 240 / 2018
Fundamenta Mathematicae 240 (2018), 161-174 MSC: 54D05, 54F15, 54F50, 54E50, 54G20. DOI: 10.4064/fm378-3-2017 Published online: 4 August 2017
A connected topological space is said to be widely-connected if each of its non-degenerate connected subsets is dense in the entire space. The object of this paper is the construction of widely-connected subsets of the plane. We give a completely metrizable example that answers a question of Paul Erdős and Howard Cook, while a similar example answers a question of Jerzy Mioduszewski.