On Liggett's approximation of functions in $C(X)$. Liggett on his 2005 book Interacting Particle Systems on page 20
 defines the state space  a  state for a particle system $X = W^S$  where one should keep in mind the case where $S = \Bbb{Z}^d$ and $W = \{0,1\}$.
Then he gives a concrete way to construct approximation of functions in $C(X)$:

My question is about the last line. When Liggett writes $\zeta \in W^T$ shouldn't it be instead $W^S$ since in this case $f_T$ and $f$ will have the same domain, otherwise it makes no sense to say that $f_T$ approximates $f$.
Is this a typo or am I missing something?
 A: I see it on page 22 in the 1985 edition. I do think the text is correct. If it is an error though, Liggett should be notified as I don't see it on the errata posted on his website.
The reason I think it's correct: For any $\xi,\zeta\in W^S$ that are identical on $T$, $f_T(\xi)=f_T(\zeta)$ since $f_T$ only sees a fixed $\eta$ outside of $T$ (I would have preferred it be called $f_{T,\eta}$).
So, yes, the domain of $f_T$ is $W^S$, but it is only supported on or can only resolve $W^T$ in a sense. So $\xi\in W^T$ is all we need to evaluate $f_T(\xi)$ (i.e. we just need to know $\xi$ restricted to $T$).
In some sense, it's cleaner they way he has done it, because it reinforces the fact that $f_T(\xi)=f_T(\zeta)$ when the configurations are identical on $T$.
Other thoughts: When he's states that $f_T(\zeta)=f(\eta^\zeta)$ for $\zeta\in W^T$, we could think of it as operating on the equivalence class $\zeta=\{\eta\in W^S \mid \eta(i)=\zeta(i), i=x,y\}$.
Alternatively we could realize that $f_T$ is defined outside of $T$ completely by $f$ and $\eta$, so it is only necessary to define $f_T$ on $W^T$, otherwise it's somewhat redundant. 
