First of all, I review some terms and notations. Let set of all ends of graph $G$ be $\Omega(G)$.
For every end $\omega$ and every finite set $S\subseteq V(G)$, there is a unique component $C(S, \omega)$ of $G−S$ that contains rays from $\omega$. Let $\Omega(S, \omega)=:\{\omega'\in \Omega|C(S,\omega')=C(S, \omega)\}$. For every $\epsilon>0$, write $E(S, \omega)$ for the set of all inner points of $S–C(S, \omega)$ edges at distance less than $\epsilon$ from their end point in $C(S, \omega)$.
Now, I am looking an example of infinite graph such that three sets $C(S, \omega)$, $\Omega(S, \omega)$ and $E(S, \omega)$ are distinct i.e $C(S, \omega)\neq\Omega(S, \omega)$ and $C(S, \omega)\neq E(S, \omega)$ and $\Omega(S, \omega)\neq E(S, \omega)$.
Note that $\Omega(G)$ is set of equivalence class of rays. So for every graph infinite graph with at least one ray, the sets $C(S, \omega)$ and $\Omega(S, \omega)$ are distinct. it is same for $\Omega(S, \omega)$ and $E(S, \omega)$. The set $E(S,\omega)$ is inner point of $S–C(S, \omega)$. Thus for every infinite graph, they are distinct.