Hindman spaces

Menachem Kojman

doi:10.1090/S0002-9939-01-06238-4

Hindman spaces

Menachem Kojman

Department of Mathematics

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

10 Scopus citations

Abstract

A topological space X is Hindman if for every sequence (x_n)_n in X there exists an infinite D ⊆N so that the sequence (x_n)_n∈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: (*) The closure of every countable set in X is compact and first-countable. 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.

Original language	English
Pages (from-to)	1597-1602
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	130
Issue number	6
DOIs	https://doi.org/10.1090/S0002-9939-01-06238-4
State	Published - 1 Jan 2002

Keywords

Compactification
Converging sequence
Finite sums
Hindman's theorem
Nonmetrizable topological spaces

ASJC Scopus subject areas

Mathematics (all)
Applied Mathematics

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10.1090/S0002-9939-01-06238-4

Cite this

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AB - A topological space X is Hindman if for every sequence (xn)n in X there exists an infinite D ⊆N so that the sequence (xn)n∈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: (*) The closure of every countable set in X is compact and first-countable. 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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Hindman spaces

Abstract

Keywords

ASJC Scopus subject areas

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