A Family of Mutually Nonhomeomorphic Separable Contractible 2-Manifolds of Cardinality $2^{2^{{\aleph }_0}}$

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.