Hindman spaces

A topological space X X is Hindman if for every sequence ( x n ) n (x_n)_n in X X there exists an infinite D ⊆ N D\subseteq \mathbb {N} so that the sequence ( x n ) n ∈ F S ( D ) (x_n)_{n\in FS(D)} , indexed by all finite sums over D D , is IP-converging in X 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 X that satisfies the following condition is Hindman: \[ ( ∗ )\quad The closure of every countable set in  X  is compact and first-countable.\quad  \text {($*$)\quad The closure of every countable set in $X$ is compact and first-countable.\quad } \] 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 ] [0,1] has an IP-converging subsequences.