Algebraic and topological properties of some sets in $l_1$

Abstract

For a sequence $x\in l_1\setminus c_{00}$, one can consider the set $E(x)$ of all subsums of series ${\sum }_{n=1}^{\infty }x(n)$. Guthrie and Nymann proved that $E(x)$ is one of the following types of sets: (I) a finite union of closed intervals; (C) homeomorphic to the Cantor set; (MC) homeomorphic to the set $T$ of subsums of ${\sum }_{n=1}^{\infty }b(n)$ where $b(2n-1)=3/4^n$ and $b(2n)=2/4^n$. By $I$, $C$ and $MC$ we denote the sets of all sequences $x\in l_1\setminus c_{00}$, such that $E(x)$ has the corresponding property. In this note we show that $I$ and $C$ are strongly $\mathfrak{c}$-algebrable and $MC$ is $\mathfrak{c}$-lineable. We show that $C$ is a dense $G_{\delta }$-set in $l_1$ and $I$ is a true $F_{\sigma }$-set. Finally we show that $I$ is spaceable while $C$ is not spaceable.

15 pages