Algebraic and topological properties of some sets in $l_1$
  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. 
T. Banakh, A. Bartoszewicz, Sz. Glab, E. Szymonik
math.GN math.FA