A shift space $(X, \sigma)$ is a *coded system* if there exist a countable collection of finite words $(\omega^n)\_{n \in \mathbb{N}}$, called generators, such that $X$ is the closure of the set of sequences obtained by freely concatenating the generators. 


In Lind and Marcus book *An introduction to symbolic dynamics and coding,* page 451 there is stated a nice topological equivalence: $(X,\sigma)$ is a coded system if and only if $X$ contain an increasing sequence of transitive subshifts of finite type whose union is dense in $X$. 


It is mentioned that this equivalence is due to Krieger and it is cited on Blanchard and Hansel paper *Sofic constant-to-one extensions of subshifts of finite type*, Proc. Amer.Math. Soc. **112** (1991), 259-265. The given reference is a personal communication between W. Krieger and the authors. 


I would like to know if there exist a reference where I can find the proof of this statement.



Coded Systems and dense subsets
In case the book referred to in Doug Lind's comment didn't have what you're looking for, a proof of this statement can be found in Section 2 of [this paper](http://www.ams.org/mathscinet-getitem?mr=1869067):


Doris Fiebig and Ulf-Rainer Fiebig, [Invariants for subshifts via nested sequences of shifts of finite type](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=90747), *Ergodic Theory and Dynamical Systems* **21** (2001), pp 1731-1758.


There are also some other references there, including an article by Krieger with the proof, and another paper by Fiebig and Fiebig with lots more information on coded systems.


