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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.14371 |
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| _version_ | 1866918250090070016 |
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| author | Kuba, Gerald |
| author_facet | Kuba, Gerald |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting continua Kuba, Gerald General Topology 54F05, 54F15 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. |
| title | Counting continua |
| topic | General Topology 54F05, 54F15 |
| url | https://arxiv.org/abs/2512.14371 |