Widely-connected sets in the bucket-handle continuum

TL;DR

This paper constructs examples of widely-connected subsets in the plane, answering longstanding questions by Erdős, Cook, and Mioduszewski about their existence and properties.

Contribution

It provides the first metrizable examples of widely-connected sets in the plane, advancing understanding of their topological structure.

Findings

Constructed a metrizable widely-connected subset of the plane.

Answered questions posed by Erdős and Cook about such sets.

Provided a similar example addressing Mioduszewski's question.

Abstract

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 Erd\H{o}s and Howard Cook, while a similar example answers a question of Jerzy Mioduszewski.