This is an announcement for the paper "Algebraic and topological properties of some sets in $l_1$" by T.Banakh, A.Bartoszewicz, Sz.Glab, and E.Szymonik.

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.

Archive classification: math.GN math.FA

Mathematics Subject Classification: 40A05, 15A03

Remarks: 15 pages

Submitted from: tbanakh@yahoo.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1208.3058

or

http://arXiv.org/abs/1208.3058