Intersections of ${\ell }^p$ spaces in the Borel hierarchy

Abstract

We show that if $Y$ is one of the spaces ${\ell }^q$, $c_0$, ${\ell }^{\infty }$ or ${\bigcap }_{p>b}{\ell }^p$ where $0<q,b<\infty$, and the Fréchet space ${\bigcap }_{p>a}{\ell }^p$ is contained in $Y$ properly, then ${\bigcap }_{p>a}{\ell }^p$ first shows up in the Borel hierarchy of $Y$ at the multiplicative class of the third level. In particular ${\bigcap }_{p>a}{\ell }^p$ 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.

12 pages, 2 figures

References in corpus (1)

• A project about chains of spaces, regarding topological and algebraic genericity and spaceability