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Hindman spaces

Author: Menachem Kojman
Journal: Proc. Amer. Math. Soc. 130 (2002), 1597-1602
MSC (1991): Primary 05C55, 54F65; Secondary 04A20, 11P99, 26A40
Published electronically: December 20, 2001
MathSciNet review: 1887003
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Abstract | References | Similar Articles | Additional Information

Abstract: A topological space $X$ is Hindman if for every sequence $(x_n)_n$ in $X$ there exists an infinite $D\subseteq \mathbb{N}$ so that the sequence $(x_n)_{n\in FS(D)}$, indexed by all finite sums over $D$, is IP-converging in $X$. Not all sequentially compact spaces are Hindman. The product of two Hindman spaces is Hindman.

Furstenberg and Weiss proved that all compact metric spaces are Hindman. We show that every Hausdorff space $X$ that satisfies the following condition is Hindman:

\begin{displaymath}\text {($*$ )\quad The closure of every countable set in $X$\space is compact and first-countable.\quad} \end{displaymath}

Consequently, there exist nonmetrizable and noncompact Hindman spaces. The following is a particular consequence of the main result: every bounded sequence of monotone (not necessarily continuous) real functions on $[0,1]$ has an IP-converging subsequences.

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Additional Information

Menachem Kojman
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, 84105, Israel

Keywords: Hindman's theorem, converging sequence, compactification, finite sums, nonmetrizable topological spaces
Received by editor(s): October 2, 2000
Received by editor(s) in revised form: December 20, 2000
Published electronically: December 20, 2001
Communicated by: Alan Dow
Article copyright: © Copyright 2001 American Mathematical Society

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