Van der Waerden spaces
Author:
Menachem Kojman
Journal:
Proc. Amer. Math. Soc. 130 (2002), 631635
MSC (2000):
Primary 05C55, 54F65; Secondary 03E05, 11P99, 26A48
Published electronically:
August 28, 2001
MathSciNet review:
1866012
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: A topological space is van der Waerden if for every sequence in there exists a converging subsequence so that 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 is sufficent for to be van der Waerden: 
 The closure of every countable set in is compact and firstcountable.
A Hausdorff space that satisfies satisfies, in fact, a stronger property: for every sequence in : 
 There exists so that is converging, and contains arbitrarily long finite arithmetic progressions and sets of the form for arbitrarily large finite sets .
There are nonmetrizable and noncompact spaces which satisfy . In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on satisfy .
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Additional Information
Menachem Kojman
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
Email:
kojman@math.bgu.ac.il
DOI:
http://dx.doi.org/10.1090/S0002993901061160
PII:
S 00029939(01)061160
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
