Moore's theorem

TL;DR

This paper reviews a simplified proof of Moore's theorem, which states that certain quotient spaces of the 2-sphere are homeomorphic to the 2-sphere, aiding applications in complex dynamics.

Contribution

It provides a clearer, more accessible proof of Moore's theorem using a simplified topological theory, facilitating its application in complex dynamics.

Findings

Proof of Moore's theorem for the 2-sphere

Simplified topological approach used in the proof

Enhanced clarity for applications in complex dynamics

Abstract

In this (mostly expository) paper, we review a proof of the following old theorem of R.L. Moore: for a closed equivalence relation on the 2-sphere such that all equivalence classes are connected and non-separating, and not all points are equivalent, the quotient space is homeomorphic to the 2-sphere. The proof uses a general topological theory close to but simpler than an original theory of Moore. The exposition is organized so that to make applications of Moore's theory (not only Moore's theorem) in complex dynamics easier, although no dynamical applications are mentioned here.