  Given a finite alphabet $\Sigma$ and a right-infinite word $w$ over the
alphabet $\Sigma$, we construct a topological space ${\rm Rec}(w)$ consisting
of all right-infinite recurrent words whose factors are all factors of $w$,
where we work up to an equivalence in which two words are equivalent if they
have the exact same set of factors (finite contiguous subwords). We show that
${\rm Rec}(w)$ can be endowed with a natural topology and we show that if $w$
is word of linear factor complexity then ${\rm Rec}(w)$ is a finite topological
space. In addition, we note that there are examples which show that if
$f:\mathbb{N}\to \mathbb{N}$ is a function that tends to infinity as $n\to
\infty$ then there is a word whose factor complexity function is ${\rm
O}(nf(n))$ such that ${\rm Rec}(w)$ is an infinite set. Finally, we pose a
realization problem: which finite topological spaces can arise as ${\rm
Rec}(w)$ for a word of linear factor complexity?
