Which connected spaces have a quotient homeomorphic to an arc

TL;DR

The paper explores the conjecture that infinite connected normal Hausdorff spaces can have quotients homeomorphic to an arc, confirming it under certain conditions and discussing challenges in the general case.

Contribution

It introduces and examines the conjecture about quotients of connected spaces being homeomorphic to an arc, providing partial results and highlighting difficulties.

Findings

Confirmed the conjecture for compact or locally connected spaces

Identified challenges in extending the result to all infinite connected normal Hausdorff spaces

Analyzed specific plane subsets to illustrate the complexities

Abstract

We announce and examine the conjecture that each infinite connected normal Hausdorff space has a quotient homeomorphic to the unit interval, shown to be true with the additional assumption of compactness or local connectedness. Some connected subsets of the plane are considered to show the difficulties involved in developing a general argument.