A well known result in Symbolic Dynamics asserts that every **two-sided** subshift on a finite alphabet necessarily consists of all doubly infinite words not containing any finite word from a given set of "forbidden" words (Proposition 1.3.4, D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge Univ. Press, 1999). This result is also true for **one-sided** subshifts (essentially the same proof works) but after researching a lot, I cannot find a direct reference to it, namely one that refers explicitly to **one-sided** subshifts. Can anyone help me find such a reference?



Reference for one-sided subshifts
My friend Marcelo Sobottka came up with the reference:



> 
> W. Ott, M. Tomforde, P. N. Willis; One-sided shift spaces over
>  infinite alphabets, NYJM Monographs (5) 2014.
> 
> 
> 


which answers my question in:



> 
> Theorem 3.16. A subset $X \subseteq \Sigma\_\mathcal A$ is a shift space if and only if $X = X\_\mathcal F$ for some subset $\mathcal F \subseteq \Sigma^{fin}\_\mathcal A$.
> 
> 
> 


This is a bit of an overkill since this result deals with shift spaces over **infinite alphabets** but, specializing the above result to the finite alphabet case, it makes me 100% satisfied! 


