Counting continua

For infinite cardinals $κ,λ$ let $C(κ,λ)$ denote the class of all compact Hausdorff spaces of weight $κ$ and size $λ$. So $C(κ,λ)=\emptyset$ if $κ>λ$ or $λ>2^κ$. If F is a class of pairwise non-homeomorphic spaces in $C(κ,λ)$ then F is a set of size not greater than $2^κ$. For every infinite cardinal $κ$ we construct $2^κ$ pairwise non-embeddable pathwise connected spaces in $C(κ,λ)$ for $λ=\max{2^{\aleph_0},κ}$ and for $λ=\exp\log(κ^+)$. (If $κ$ is a strong limit then $\exp\log(κ^+)=2^κ$.) Additionally, for all infinite cardinals $κ,μ$ with $μ\leqκ$ we construct $2^κ$ pairwise non-embeddable connected spaces in $C(κ,κ^μ)$. Furthermore, for $κ=λ=2^θ$ with arbitrary $θ$ and for certain other pairs $κ,λ$ we construct $2^κ$ pairwise non-embeddable connected, linearly ordered spaces $X\in C(κ,λ)$ such that $Y\in C(κ,λ)$ 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 $λ$ is of countable cofinality and either $κ=λ$ or $λ$ is a strong limit.

💡 Research Summary

The paper investigates the landscape of compact Hausdorff spaces whose weight and cardinality are prescribed by two infinite cardinals κ and λ. The class C(κ,λ) consists of all compact Hausdorff spaces of weight κ and size λ; it is empty when κ>λ or λ>2^κ. A classical counting argument shows that any family of pairwise non‑homeomorphic members of C(κ,λ) can have at most 2^κ members, because every normal space of weight κ embeds into the Hilbert cube