Van der Waerden spaces and Hindman spaces are not the same

Menachem Kojman; Saharon Shelah

doi:10.1090/S0002-9939-02-06916-2

Van der Waerden spaces and Hindman spaces are not the same

Menachem Kojman, Saharon Shelah

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

A Hausdorff topological space X is van der Waerden if for every sequence (x_n)_n∈w in X there is a converging subsequence (x_n)_n∈A where A ⊆ w contains arithmetic progressions of all finite lengths. A Hausdorff topological space X is Hindman if for every sequence (x_n)_n∈w in X there is an IP-converging subsequence (x_n)_n∈FS(B) for some infinite B ⊆ w. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

Original language	English
Pages (from-to)	1619-1622
Number of pages	4
Journal	Proceedings of the American Mathematical Society
Volume	131
Issue number	5
DOIs	https://doi.org/10.1090/S0002-9939-02-06916-2
State	Published - 1 May 2003

ASJC Scopus subject areas

Mathematics (all)
Applied Mathematics

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10.1090/S0002-9939-02-06916-2

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abstract = "A Hausdorff topological space X is van der Waerden if for every sequence (xn)n∈w in X there is a converging subsequence (xn)n∈A where A ⊆ w contains arithmetic progressions of all finite lengths. A Hausdorff topological space X is Hindman if for every sequence (xn)n∈w in X there is an IP-converging subsequence (xn)n∈FS(B) for some infinite B ⊆ w. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.",

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AB - A Hausdorff topological space X is van der Waerden if for every sequence (xn)n∈w in X there is a converging subsequence (xn)n∈A where A ⊆ w contains arithmetic progressions of all finite lengths. A Hausdorff topological space X is Hindman if for every sequence (xn)n∈w in X there is an IP-converging subsequence (xn)n∈FS(B) for some infinite B ⊆ w. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

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Van der Waerden spaces and Hindman spaces are not the same

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