Counting continua

TL;DR

This paper constructs large families of pairwise non-embeddable compact Hausdorff spaces with specified weight and size, exploring their properties and limitations in the context of infinite cardinals and connectedness.

Contribution

It introduces new constructions of vast families of non-embeddable spaces within classes defined by cardinality and connectedness, and analyzes their embedding properties and limitations.

Findings

Constructed 2^κ non-embeddable pathwise connected spaces for various λ.

Built 2^κ non-embeddable connected spaces with specific cardinalities.

Identified conditions where such spaces cannot exist, especially with countable cofinality.

Abstract

For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a class of pairwise non-homeomorphic spaces in $C(\kappa,\lambda)$ then F is a set of size not greater than $2^\kappa$. For every infinite cardinal $\kappa$ we construct $2^\kappa$ pairwise non-embeddable pathwise connected spaces in $C(\kappa,\lambda)$ for $\lambda=\max\{2^{\aleph_0},\kappa\}$ and for $\lambda=\exp\log(\kappa^+)$. (If $\kappa$ is a strong limit then $\exp\log(\kappa^+)=2^\kappa$.) Additionally, for all infinite cardinals $\kappa,\mu$ with $\mu\leq\kappa$ we construct $2^\kappa$ pairwise non-embeddable connected spaces in $C(\kappa,\kappa^\mu)$. Furthermore, for $\kappa=\lambda=2^\theta$ with arbitrary $\theta$ and for certain other pairs $\kappa,\lambda$ we construct $2^{\kappa}$ pairwise non-embeddable connected, linearly ordered spaces $X\in C(\kappa,\lambda)$ such that $Y\in C(\kappa,\lambda)$ whenever $Y$ is an infinite compact and connected subspace of $X$. On the other hand we prove that there is no space $X$ with this property if $\lambda$ is of countable cofinality and either $\kappa=\lambda$ or $\lambda$ is a strong limit.