Topological complexity of ideal limit points
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913434316046336 |
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| author | Balcerzak, Marek Glab, Szymon Leonetti, Paolo |
| author_facet | Balcerzak, Marek Glab, Szymon Leonetti, Paolo |
| contents | Given an ideal $\mathcal{I}$ on the nonnegative integers $ω$ and a Polish space $X$, let $\mathscr{L}(\mathcal{I})$ be the family of subsets $S\subseteq X$ such that $S$ is the set of $\mathcal{I}$-limit points of some sequence taking values in $X$. First, we show that $\mathscr{L}(\mathcal{I})$ may attain arbitrarily large Borel complexity. Second, we prove that if $\mathcal{I}$ is a $G_{δσ}$-ideal then all elements of $\mathscr{L}(\mathcal{I})$ are closed. Third, we show that if $\mathcal{I}$ is a simply coanalytic ideal and $X$ is first countable, then every element of $\mathscr{L}(\mathcal{I})$ is simply analytic. Lastly, we studied certain structural properties and the topological complexity of minimal ideals $\mathcal{I}$ for which $\mathscr{L}(\mathcal{I})$ contains a given set. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_12160 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological complexity of ideal limit points Balcerzak, Marek Glab, Szymon Leonetti, Paolo General Topology Classical Analysis and ODEs Functional Analysis Given an ideal $\mathcal{I}$ on the nonnegative integers $ω$ and a Polish space $X$, let $\mathscr{L}(\mathcal{I})$ be the family of subsets $S\subseteq X$ such that $S$ is the set of $\mathcal{I}$-limit points of some sequence taking values in $X$. First, we show that $\mathscr{L}(\mathcal{I})$ may attain arbitrarily large Borel complexity. Second, we prove that if $\mathcal{I}$ is a $G_{δσ}$-ideal then all elements of $\mathscr{L}(\mathcal{I})$ are closed. Third, we show that if $\mathcal{I}$ is a simply coanalytic ideal and $X$ is first countable, then every element of $\mathscr{L}(\mathcal{I})$ is simply analytic. Lastly, we studied certain structural properties and the topological complexity of minimal ideals $\mathcal{I}$ for which $\mathscr{L}(\mathcal{I})$ contains a given set. |
| title | Topological complexity of ideal limit points |
| topic | General Topology Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2407.12160 |