Unboring ideals
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910770610044928 |
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| author | Kwela, Adam |
| author_facet | Kwela, Adam |
| contents | Our main object of interest is the following notion: we say that a topological space space $X$ is in FinBW($\mathcal{I}$), where $\mathcal{I}$ is an ideal on $ω$, if for each sequence $(x_n)_{n\inω}$ in $X$ one can find an $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ converges in $X$. We define an ideal $\mathcal{BI}$ which is critical for FinBW($\mathcal{I}$) in the following sense: Under CH, for every ideal $\mathcal{I}$, $\mathcal{BI}\not\leq_K\mathcal{I}$ ($\leq_K$ denotes the Katětov preorder of ideals) iff there is an uncountable separable space in FinBW($\mathcal{I}$). We show that $\mathcal{BI}\not\leq_K\mathcal{I}$ and $ω_1$ with the order topology is in FinBW($\mathcal{I}$), for all $\bf{Π^0_4}$ ideals $\mathcal{I}$. We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is nonempty: we prove under MA($σ$-centered) that for $\bf{Π^0_4}$ ideals $\mathcal{I}$ and $\mathcal{J}$ this is equivalent to $\mathcal{J}\not\leq_K\mathcal{I}$. Moreover, answering in negative a question of M. Hrušák and D. Meza-Alcántara, we show that the ideal $\text{Fin}\times\text{Fin}$ is not critical among Borel ideals for extendability to a $\bf{Π^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_17166 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Unboring ideals Kwela, Adam General Topology Logic Primary: 03E05. Secondary: 03E15, 03E35, 26A03, 40A05, 54A20, 54H05 Our main object of interest is the following notion: we say that a topological space space $X$ is in FinBW($\mathcal{I}$), where $\mathcal{I}$ is an ideal on $ω$, if for each sequence $(x_n)_{n\inω}$ in $X$ one can find an $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ converges in $X$. We define an ideal $\mathcal{BI}$ which is critical for FinBW($\mathcal{I}$) in the following sense: Under CH, for every ideal $\mathcal{I}$, $\mathcal{BI}\not\leq_K\mathcal{I}$ ($\leq_K$ denotes the Katětov preorder of ideals) iff there is an uncountable separable space in FinBW($\mathcal{I}$). We show that $\mathcal{BI}\not\leq_K\mathcal{I}$ and $ω_1$ with the order topology is in FinBW($\mathcal{I}$), for all $\bf{Π^0_4}$ ideals $\mathcal{I}$. We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is nonempty: we prove under MA($σ$-centered) that for $\bf{Π^0_4}$ ideals $\mathcal{I}$ and $\mathcal{J}$ this is equivalent to $\mathcal{J}\not\leq_K\mathcal{I}$. Moreover, answering in negative a question of M. Hrušák and D. Meza-Alcántara, we show that the ideal $\text{Fin}\times\text{Fin}$ is not critical among Borel ideals for extendability to a $\bf{Π^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals. |
| title | Unboring ideals |
| topic | General Topology Logic Primary: 03E05. Secondary: 03E15, 03E35, 26A03, 40A05, 54A20, 54H05 |
| url | https://arxiv.org/abs/2103.17166 |