Algebraic and topological properties of some sets in $\ell_1$
Volume 129 / 2012
Abstract
For a sequence $x \in\ell_1 \setminus c_{00}$, one can consider the set $E(x)$ of all subsums of the series $\sum_{n=1}^{\infty} x(n)$. Guthrie and Nymann proved that $E(x)$ is one of the following types of sets: $(\mathcal{I})$ a finite union of closed intervals; $(\mathcal{C})$ homeomorphic to the Cantor set; $(\mathcal{MC})$ homeomorphic to the set $T$ of subsums of $\sum_{n=1}^\infty b(n)$ where $b(2n-1) = 3/4^n$ and $b(2n) = 2/4^n$. Denote by $\mathcal I$, $\mathcal C$ and $\mathcal{MC}$ the sets of all sequences $x \in\ell_1 \setminus c_{00}$ such that $E(x)$ has the property ($\mathcal I$), ($\mathcal C$) and ($\mathcal{MC}$), respectively. We show that $\mathcal I$ and $\mathcal C$ are strongly $\mathfrak{c}$-algebrable and $\mathcal{MC}$ is $\mathfrak{c}$-lineable. We also show that $\mathcal C$ is a dense $\mathcal G_\delta$-set in $\ell_1$ and $\mathcal I$ is a true $\mathcal F_\sigma$-set. Finally we show that $\mathcal I$ is spaceable while $\mathcal C$ is not.
