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Keywords:
ultrafilter; $0$-point; summable ideal; linked family
ultrafilter; $0$-point; summable ideal; linked family
Summary:
\font\tenscr = rsfs10 \font\sevenscr = rsfs7 \font\fivescr = rsfs5 \textfont14=\tenscr \scriptfont14=\sevenscr \scriptscriptfont14=\fivescr \def\scr{\fam14} We construct in ZFC an ultrafilter $\scr U \in \Bbb N^{\ast}$ such that for every one-to-one function $f : \Bbb N\rightarrow \Bbb N$ there exists $U\in \scr U$ with $f[U]$ in the summable ideal, i.e. the sum of reciprocals of its elements converges. This strengthens Gryzlov's result concerning the existence of $0$-points.
References:
[1] Flašková J.: Ultrafilters and two ideals on $ømega $. in: WDS'05 Proceedings of Contributed Papers: Part I - Mathematics and Computer Sciences (ed. Jana Šafránková), Prague, Matfyzpress, 2005, pp.78-83.
[2] Gryzlov A.: Some types of points in $N^{*}$. in: Proceedings of the 12th Winter School on Abstract Analysis (Srní, 1984), Rend. Circ. Mat. Palermo (2) Suppl. No. 6, 1984, pp.137-138. MR 0782711 | Zbl 0566.54011
[3] Gryzlov A.A.: On theory of the space $\beta \Bbb N$. General Topology (Russian), 166, Moskov. Gos. Univ., Moscow, 1986, pp.20-34. MR 1080755
[4] Hart K.P.: Notes taken at Winter School 1984, Srní, handwritten notes.
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