Editing Potts model (section)

=== Topology of the space of states ===
Let ''Q'' = {1, ..., ''q''} be a finite set of symbols, and let
: <math>Q^\mathbf{Z}=\{ s=(\ldots,s_{-1},s_0,s_1,\ldots) :  s_k \in Q \; \forall k \in \mathbf{Z} \}</math>
be the set of all bi-infinite strings of values from the set ''Q''. This set is called a [[full shift]]. For defining the Potts model, either this whole space, or a certain subset of it, a [[subshift of finite type]], may be used. Shifts get this name because there exists a natural operator on this space, the [[shift operator]] τ : ''Q''<sup>'''Z'''</sup> → ''Q''<sup>'''Z'''</sup>, acting as
: <math>\tau (s)_k = s_{k+1}</math>

This set has a natural [[product topology]]; the [[base (topology)|base]] for this topology are the [[cylinder set]]s
: <math>C_m[\xi_0, \ldots, \xi_k]= \{s \in Q^\mathbf{Z} : s_m = \xi_0, \ldots ,s_{m+k} = \xi_k \}</math>
that is, the set of all possible strings where ''k''+1 spins match up exactly to a given, specific set of values ξ<sub>0</sub>, ..., ξ<sub>''k''</sub>. Explicit representations for the cylinder sets can be gotten by noting that the string of values corresponds to a [[p-adic number|''q''-adic number]], however the natural topology of the q-adic numbers is finer than the above product topology.