A two-sided shift space, also called a subshift, is a closed set 
in a full shift 
such that 
, where 
is a finite alphabet and the
left shift is
 |
The product topology is formed from the discrete topology on 
.
The restriction of 
to 
defines a discrete dynamical
system.
Equivalently, a two-sided shift space consists of the sequences that avoid a specified collection 
of forbidden finite words,
 |
The collection may be finite or infinite. For example, forbidding the word 
over the alphabet 
permits exactly the bi-infinite
sequences with no adjacent 1s (Lind and Marcus 2021,
pp. 5-8).
For one-sided sequences, indexed by the nonnegative integers, one commonly requires only 
in addition to being a nonempty closed
set (Kůrka 2009). The shift need not be onto.
For example, if 
denotes the all-zero sequence, then 
satisfies this definition but 
.
