The Nadler-Quinn problem on accessible points of arc-like continua

TL;DR

This paper proves that every point in an arc-like continuum can be made accessible through a plane embedding, solving a longstanding problem in continuum theory and developing new theories of interval map factorizations.

Contribution

It introduces the theories of truncations and contour factorizations of interval maps to solve the Nadler-Quinn problem.

Findings

Every point in an arc-like continuum can be accessible in some plane embedding.

Answers the Nadler-Quinn problem posed in 1972.

Provides new insights into inequivalent embeddings of indecomposable arc-like continua.

Abstract

We show that if $X$ is an arc-like continuum, then for every point $x \in X$ there is a plane embedding of $X$ in which $x$ is an accessible point. This answers a question posed by Nadler in 1972, which has become known as the Nadler-Quinn problem in continuum theory. Towards this end, we develop the theories of truncations and contour factorizations of interval maps. As a corollary, we answer a question of Mayer from 1982 about inequivalent plane embeddings of indecomposable arc-like continua.