Intersections of $\ell^p$ spaces in the Borel hierarchy
| Autor: | Gregoriades, Vassilios |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We show that if $Y$ is one of the spaces $\ell^q$, $c_0$, $\ell^\infty$ or ${\textstyle \bigcap_{p > b}} \ell^p$ where $0 < q,b < \infty$, and the Fr\'{e}chet space $\textstyle \bigcap_{p > a} \ell^p$ is contained in $Y$ properly, then $\textstyle \bigcap_{p > a} \ell^p$ first shows up in the Borel hierarchy of $Y$ at the multiplicative class of the third level. In particular $\textstyle \bigcap_{p > a} \ell^p$ is neither an $F_\sigma$ nor a $G_\delta$ subset of $Y$. This answers a question by Nestoridis. This result provides a natural example of a set in the third level of the Borel hierarchy and with its help we also give some examples in the fourth level. Comment: 12 pages, 2 figures |
| Databáze: | arXiv |
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