Counting spaces of functions on separable compact lines
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908887131619328 |
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| author | Korpalski, Maciej Koszmider, Piotr Marciszewski, Witold |
| author_facet | Korpalski, Maciej Koszmider, Piotr Marciszewski, Witold |
| contents | We investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces $C(K)$ of continuous real-valued functions on a compact space $K$, equipped with the supremum norm: Let $\mathcal{K}$ be a class of compact spaces. How many isomorphism types of Banach spaces $C(K)$ are there, for $K\in \mathcal{K}$?
We prove that for any uncountable regular cardinal number $κ$, there exist exactly $2^κ$ isomorphism types of spaces $C(K)$ for compact spaces of weight $κ$.
We show that, for the class $\mathcal{L}_{ω_1}$ of separable compact linearly ordered spaces of weight $ω_1$, the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are $2^{ω_1}$ isomorphism types of $C(L)$, for $L\in \mathcal{L_{ω_1}}$, and assuming a certain axiom proposed by Baumgartner, there is only one type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09143 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting spaces of functions on separable compact lines Korpalski, Maciej Koszmider, Piotr Marciszewski, Witold Functional Analysis General Topology Logic 03E35, 03E65, 03E75, 46B03, 46E15, 54F05 We investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces $C(K)$ of continuous real-valued functions on a compact space $K$, equipped with the supremum norm: Let $\mathcal{K}$ be a class of compact spaces. How many isomorphism types of Banach spaces $C(K)$ are there, for $K\in \mathcal{K}$? We prove that for any uncountable regular cardinal number $κ$, there exist exactly $2^κ$ isomorphism types of spaces $C(K)$ for compact spaces of weight $κ$. We show that, for the class $\mathcal{L}_{ω_1}$ of separable compact linearly ordered spaces of weight $ω_1$, the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are $2^{ω_1}$ isomorphism types of $C(L)$, for $L\in \mathcal{L_{ω_1}}$, and assuming a certain axiom proposed by Baumgartner, there is only one type. |
| title | Counting spaces of functions on separable compact lines |
| topic | Functional Analysis General Topology Logic 03E35, 03E65, 03E75, 46B03, 46E15, 54F05 |
| url | https://arxiv.org/abs/2602.09143 |