Topological invariants for words of linear factor complexity

Given a finite alphabet $\mathrm{\Sigma }$ and a right-infinite word $w$ over the alphabet $\mathrm{\Sigma }$, we construct a topological space $\mathrm{R}\mathrm{e}\mathrm{c}(w)$ 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 $\mathrm{R}\mathrm{e}\mathrm{c}(w)$ can be endowed with a natural topology and we show that if $w$ is word of linear factor complexity then $\mathrm{R}\mathrm{e}\mathrm{c}(w)$ 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 $n\to \infty$thenthereisawordwhosefactorcomplexityfunctionis${\rm O}(nf(n))$suchthat${\rm Rec}(w)$ 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?