A Family of Mutually Nonhomeomorphic Separable Contractible 2-Manifolds of Cardinality $2^{2^{\aleph_0}}$

TL;DR

This paper constructs a vast family of separable, contractible 2-manifolds embedded in the Moore plane, demonstrating the existence of an enormous number of mutually nonhomeomorphic examples, especially among nonmetrizable manifolds.

Contribution

It introduces a large family of open subsets of the Moore plane, each forming a separable, contractible 2-manifold with uncountably many nonhomeomorphic classes.

Findings

Contains $2^{2^{eth_1}}$ distinct homeomorphism classes.

Includes both metrizable and nonmetrizable manifolds.

Provides explicit construction of these manifolds.

Abstract

We describe a family of open subsets $M_A$ of the Moore plane, one for each subset $A$ of $\mathbb{R}$. Each of these subsets is a separable contractible (Hausdorff) 2-manifold, which is nonmetrizable if $A$ is uncountable. The collection $\{M_A:A\subseteq\mathbb{R}\}$ contains $2^{2^{\aleph_0}}$ distinct homeomorphism classes.