All spaces of countable spread can be small
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911343708209152 |
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| author | Dow, Alan Juhász, István |
| author_facet | Dow, Alan Juhász, István |
| contents | The main result of this paper is the proof of the simultaneous consistency, modulo a weakly compact cardinal, of the equality $2^{< \mathfrak{c}} = \mathfrak{c}$ with the following property (*) of partitions of pairs of $\mathfrak{c}$:
\smallskip
(*) For any coloring (or partition) $k : [\mathfrak{c}]^2 \rightarrow 2$ either there is a homogeneous set of size $\mathfrak{c}$ in color $0$ or there is a set $S \in [\mathfrak{c}]^\mathfrak{c}$ such that for every countable $A \subset S$ there is $β\in \mathfrak{c}$ for which $A \subset β$ and $k(\{α, β\}) = 1$ for all $α\in A$.
\smallskip
(*) plus $2^{< \mathfrak{c}} = \mathfrak{c}$ together then imply that for every topological space $X$ of countable
spread, i.e. not containing any uncountable discrete subset,
$|X| \le \mathfrak{c}$ if it is
Hausdorff and
$o(X) = \mathfrak c$ if it is also infinite and regular. Here $o(X)$ denotes the number of all open subsets of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | All spaces of countable spread can be small Dow, Alan Juhász, István General Topology Logic 54A25 The main result of this paper is the proof of the simultaneous consistency, modulo a weakly compact cardinal, of the equality $2^{< \mathfrak{c}} = \mathfrak{c}$ with the following property (*) of partitions of pairs of $\mathfrak{c}$: \smallskip (*) For any coloring (or partition) $k : [\mathfrak{c}]^2 \rightarrow 2$ either there is a homogeneous set of size $\mathfrak{c}$ in color $0$ or there is a set $S \in [\mathfrak{c}]^\mathfrak{c}$ such that for every countable $A \subset S$ there is $β\in \mathfrak{c}$ for which $A \subset β$ and $k(\{α, β\}) = 1$ for all $α\in A$. \smallskip (*) plus $2^{< \mathfrak{c}} = \mathfrak{c}$ together then imply that for every topological space $X$ of countable spread, i.e. not containing any uncountable discrete subset, $|X| \le \mathfrak{c}$ if it is Hausdorff and $o(X) = \mathfrak c$ if it is also infinite and regular. Here $o(X)$ denotes the number of all open subsets of $X$. |
| title | All spaces of countable spread can be small |
| topic | General Topology Logic 54A25 |
| url | https://arxiv.org/abs/2512.23544 |