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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.12187 |
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| _version_ | 1866912830124457984 |
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| author | Filipów, Rafał Kwela, Adam Leonetti, Paolo |
| author_facet | Filipów, Rafał Kwela, Adam Leonetti, Paolo |
| contents | Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq ω$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in ω}$ taking values in $X$, let $Λ_{x}(FS)$ be the set of $IP$-limit points of $x$. Also, let $Λ_{x}(\mathcal{H})$ be the set of $\mathcal{H}$-limit points of $x$, that is, the set of ordinary limits of subsequences $(x_n)_{n \in S}$ with $S\notin \mathcal{H}$. After proving that these two notions do not coincide in general, we show that both families of nonempty sets of the type $Λ_{x}(FS)$ and of the type $Λ_{x}(\mathcal{H})$ are precisely the class of nonempty analytic subsets of $X$.
An analogous result holds also for Ramsey convergence. In the proofs, we use the concept of partition regular functions introduced in J. Symb. Log. (2024) [doi:10.1017/jsl.2024.8], which provide a unified approach to these types of convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12187 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sets of Ramsey-limit points and IP-limit points Filipów, Rafał Kwela, Adam Leonetti, Paolo General Topology Dynamical Systems Logic 03E75, 05D10, 40A05, 54A20 (Primary) 03E05, 11B05, 28A05, 40A35 (Secondary) Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq ω$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in ω}$ taking values in $X$, let $Λ_{x}(FS)$ be the set of $IP$-limit points of $x$. Also, let $Λ_{x}(\mathcal{H})$ be the set of $\mathcal{H}$-limit points of $x$, that is, the set of ordinary limits of subsequences $(x_n)_{n \in S}$ with $S\notin \mathcal{H}$. After proving that these two notions do not coincide in general, we show that both families of nonempty sets of the type $Λ_{x}(FS)$ and of the type $Λ_{x}(\mathcal{H})$ are precisely the class of nonempty analytic subsets of $X$. An analogous result holds also for Ramsey convergence. In the proofs, we use the concept of partition regular functions introduced in J. Symb. Log. (2024) [doi:10.1017/jsl.2024.8], which provide a unified approach to these types of convergence. |
| title | Sets of Ramsey-limit points and IP-limit points |
| topic | General Topology Dynamical Systems Logic 03E75, 05D10, 40A05, 54A20 (Primary) 03E05, 11B05, 28A05, 40A35 (Secondary) |
| url | https://arxiv.org/abs/2601.12187 |