Van der Waerden spaces and Hindman spaces are not the same

Abstract

A Hausdorff topological space X X is van der Waerden if for every sequence ( x n ) n ∈ ω (x_n)_{n\in \omega } in X X there is a converging subsequence ( x n ) n ∈ A (x_n)_{n\in A} where A ⊆ ω A\subseteq \omega contains arithmetic progressions of all finite lengths. A Hausdorff topological space X X is Hindman if for every sequence ( x n ) n ∈ ω (x_n)_{n\in \omega } in X X there is an IP-converging subsequence ( x n ) n ∈ F S ( B ) (x_n)_{n\in FS(B)} for some infinite B ⊆ ω B\subseteq \omega . We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.