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Van der Waerden spaces

Author: Menachem Kojman
Journal: Proc. Amer. Math. Soc. 130 (2002), 631-635
MSC (2000): Primary 05C55, 54F65; Secondary 03E05, 11P99, 26A48
Published electronically: August 28, 2001
MathSciNet review: 1866012
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Abstract: A topological space $X$ is van der Waerden if for every sequence $(x_n)_n$ in $X$ there exists a converging subsequence $(x_{n_k})_k$ so that $\{{n_k}:k\in \mathbb{N}\}$ contains arbitrarily long finite arithmetic progressions. Not every sequentially compact space is van der Waerden. The product of two van der Waerden spaces is van der Waerden.

The following condition on a Hausdorff space $X$ is sufficent for $X$to be van der Waerden:

The closure of every countable set in $X$ is compact and first-countable.

A Hausdorff space $X$ that satisfies $(*)$ satisfies, in fact, a stronger property: for every sequence $(x_n)$ in $X$:

There exists $A\subseteq\mathbb N$ so that $(x_n)_{n\in A}$ is converging, and $A$ contains arbitrarily long finite arithmetic progressions and sets of the form $FS(D)$ for arbitrarily large finite sets $D$.

There are nonmetrizable and noncompact spaces which satisfy $(*)$. In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on $[0,1]$ satisfy $(\star)$.

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Additional Information

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

Keywords: van der Waerden's Theorem, converging sequence, compactification, finite sums
Received by editor(s): August 2, 2000
Received by editor(s) in revised form: August 28, 2000
Published electronically: August 28, 2001
Additional Notes: The author thanks Uri Abraham for many discussions that contributed to the development of this paper, and also thanks the referee for some constructive and helpful comments
Communicated by: Alan Dow
Article copyright: © Copyright 2001 American Mathematical Society

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