Van der Waerden spaces

A topological space $X$ is van der Waerden if for every sequence $(x_n)_n$ in $X$ there exists a converging subsequence $(x_{n_k})_k$ so that $\{{n_k}:k\in \mathbb{N}\}$ 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 $X$ is sufficent for $X$to be van der Waerden:

$(*)$

The closure of every countable set in $X$ is compact and first-countable.

A Hausdorff space $X$ that satisfies $(*)$ satisfies, in fact, a stronger property: for every sequence $(x_n)$ in $X$:

$(\star)$

There exists $A\subseteq\mathbb N$ so that $(x_n)_{n\in A}$ is converging, and $A$ contains arbitrarily long finite arithmetic progressions and sets of the form $FS(D)$ for arbitrarily large finite sets $D$.

There are nonmetrizable and noncompact spaces which satisfy $(*)$. In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on $[0,1]$ satisfy $(\star)$.