Counting continua

Abstract

For infinite cardinals $\hat{I}º,\hat{I}»$ let $C(\hat{I}º,\hat{I}»)$ denote the class of all compact Hausdorff spaces of weight $\hat{I}º$ and size $\hat{I}»$. So $C(\hat{I}º,\hat{I}»)=\varnothing$ if $\hat{I}º>\hat{I}»$ or $\hat{I}»>2^{\hat{I}}º$. If F is a class of pairwise non-homeomorphic spaces in $C(\hat{I}º,\hat{I}»)$ then F is a set of size not greater than $2^{\hat{I}}º$. For every infinite cardinal $\hat{I}º$ we construct $2^{\hat{I}}º$ pairwise non-embeddable pathwise connected spaces in $C(\hat{I}º,\hat{I}»)$ for $\hat{I}»=\max \{2^{{\aleph }_0},\hat{I}º\}$ and for $\hat{I}»=\exp \log (\hat{I}{º}^{+})$. (If $\hat{I}º$ is a strong limit then $\exp \log (\hat{I}{º}^{+})=2^{\hat{I}}º$.) Additionally, for all infinite cardinals $\hat{I}º,\hat{I}¼$ with $\hat{I}¼\le \hat{I}º$ we construct $2^{\hat{I}}º$ pairwise non-embeddable connected spaces in $C(\hat{I}º,\hat{I}{º}^{\hat{I}}¼)$. Furthermore, for $\hat{I}º=\hat{I}»=2^{\hat{I}}¸$ with arbitrary $\hat{I}¸$ and for certain other pairs $\hat{I}º,\hat{I}»$ we construct $2^{\hat{I}}º$ pairwise non-embeddable connected, linearly ordered spaces $X\in C(\hat{I}º,\hat{I}»)$ such that $Y\in C(\hat{I}º,\hat{I}»)$ 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 $\hat{I}»$ is of countable cofinality and either $\hat{I}º=\hat{I}»$ or $\hat{I}»$ is a strong limit.