Counting continua

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.