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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2603.01114 |
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| _version_ | 1866911476171669504 |
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| author | Kwela, Adam Lesner, Dorota |
| author_facet | Kwela, Adam Lesner, Dorota |
| contents | Let $\mathcal{I}$ be an ideal on $ω$ and $X$ be a topological space. A sequence $(x_n)_{n\in ω}$ in $X$ is $\mathcal{I}$-convergent if there is $x\in X$ such that $\{n\in ω:x_n\notin U\}\in\mathcal{I}$ for every open neighborhood $U$ of $x$. We examine the following variant of sequential compactness associated with $\I$: $X$ is $\mathrm{BW}(\mathcal{I})$ if for every sequence $(x_n)_{n\in ω}$ in $X$ there is $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ is $\mathcal{I}$-convergent.
We introduce a new preorder on ideals, denoted $\leq_{BW}$, such that $\mathcal{I}\leq_{BW}\mathcal{J}$ implies that every $\mathrm{BW}(\mathcal{J})$ space is $\mathrm{BW}(\mathcal{I})$. Our main result states that under CH the above implication can be reversed in the case of $\mathbf{F_σ}$ ideals $\I$ and $\J$.
We compare $\leq_{BW}$ with the Katětov order and study the relation $\leq_{BW}$ among some well-known ideals (e.g. the van der Waerden ideal $\mathcal{W}$ consisting of all subsets of $ω$ that do not contain arbitrary long finite arithmetic progressions). As a consequence, we answer two open questions posed by Filipów and Tryba in [Top. App. {\textbf{178}} (2014), 438--452] concerning comparison of $\mathrm{BW}(\mathcal{W})$ with the class of sequentially compact spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01114 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new order for ideal sequential compactness Kwela, Adam Lesner, Dorota General Topology Let $\mathcal{I}$ be an ideal on $ω$ and $X$ be a topological space. A sequence $(x_n)_{n\in ω}$ in $X$ is $\mathcal{I}$-convergent if there is $x\in X$ such that $\{n\in ω:x_n\notin U\}\in\mathcal{I}$ for every open neighborhood $U$ of $x$. We examine the following variant of sequential compactness associated with $\I$: $X$ is $\mathrm{BW}(\mathcal{I})$ if for every sequence $(x_n)_{n\in ω}$ in $X$ there is $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ is $\mathcal{I}$-convergent. We introduce a new preorder on ideals, denoted $\leq_{BW}$, such that $\mathcal{I}\leq_{BW}\mathcal{J}$ implies that every $\mathrm{BW}(\mathcal{J})$ space is $\mathrm{BW}(\mathcal{I})$. Our main result states that under CH the above implication can be reversed in the case of $\mathbf{F_σ}$ ideals $\I$ and $\J$. We compare $\leq_{BW}$ with the Katětov order and study the relation $\leq_{BW}$ among some well-known ideals (e.g. the van der Waerden ideal $\mathcal{W}$ consisting of all subsets of $ω$ that do not contain arbitrary long finite arithmetic progressions). As a consequence, we answer two open questions posed by Filipów and Tryba in [Top. App. {\textbf{178}} (2014), 438--452] concerning comparison of $\mathrm{BW}(\mathcal{W})$ with the class of sequentially compact spaces. |
| title | A new order for ideal sequential compactness |
| topic | General Topology |
| url | https://arxiv.org/abs/2603.01114 |