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

TL;DR

This paper characterizes the Borel hierarchy level of certain intersections of ^p spaces, showing they are at the third level and not simpler, thus providing natural examples of complex Borel sets.

Contribution

It identifies the precise Borel hierarchy level of intersections of ^p spaces, answering a question by Nestoridis and providing new examples in the hierarchy.

Findings

^p intersections are at the third Borel level in certain spaces.

These sets are neither _sigma nor G_delta.

Provides examples of sets in the third and fourth Borel levels.

Abstract

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.