Peano Quotients of Metric Continua

TL;DR

This paper establishes that for any nonempty compact connected metric space $M$, the congestion set $\mathscr N(M)$ (via its closure $\mathcal F$ and the complement $\mathcal O$) induces a partition $\mathscr P=\mathscr F\cup\mathscr O$ whose quotient metric space $(\mathscr Q,\nabla_{\mathscr Q})$ satisfies $\mathscr Q=\mathscr P$ and is a Peano continuum. The author shows that $\mathscr F$ is totally disconnected in the quotient and that $\nabla_{\mathscr Q}$ agrees locally with $d$ on $\mathcal O$, giving a canonical Peano quotient that retains the local metric structure away from congestion points. Special cases include when $\mathcal F=\emptyset$, where the quotient is isometric to $M$, and when $\mathcal F=M$, where the partition collapses to a singleton; for continua, the quotient remains a Peano continuum and is a continuous image of a closed interval by the Hahn–Mazurkiewicz theorem. The work also discusses generalizations to other closed sets containing congestion points, showing the quotient remains Peano under these constructions. Overall, the paper connects partition-based quotient construction with classical continuum theory to produce well-behaved Peano quotients that encapsulate the weakly locally connected structure of $M$.

Abstract

For any compact, connected metric space $(M,d)$ the set of points where $M$ is not weakly locally connected is shown to define a partition $\sP$ of $M$ for which the corresponding quotient metric space $(\sQ, \nabla_\sQ)$ is a Peano continuum with $\sQ = \sP$.

Theorems & Definitions (30)

• Remark 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Definition 2.6
• Theorem 2.7
• proof
• Remark 2.8
• Definition 2.9