Let $E := \{ p \in \mathbb{Q} : 0 < p < \sqrt 2 \}$.
(1) $\sqrt 2$ is not an inner point of $E$ because $\forall \delta \in \mathbb{R} : \sqrt 2 \not \in \bigcup_{x \in I} B(x, \delta) \subset E \subset \mathbb{Q} \subset \mathbb{R}$ and $\sqrt 2$ is not an outer point because $\forall \delta \in \mathbb{R} : \sqrt 2 \not \in \bigcup_{x\in I_{2}} B(x, \delta) \subset E^{c} \subset \mathbb{Q} \subset \mathbb{R}$. So $\sqrt 2$ is a boundary point of $\mathbb{Q}$ in $\mathbb{R}$.
(2) Notice that $0 \in \mathbb{R}$ and $0 \in \mathbb{Q}$. Since $\forall \delta > 0 : 0 \in \bigcup_{x\in I_{3}} B(x, \delta) \subset E \subset \mathbb{Q} \subset \mathbb{R}$ so $0$ is an inner point. $0$ is also an outer point because $\forall \delta > 0 : 0 \in \bigcup_{x \in I_{4}} B(x, \delta) \subset E^{c} \subset \mathbb{Q} \subset \mathbb{R}$. No boundary point.
[conlclusion, investigating]
Can you declare the borders in some other numbers such as complex numbers or some other way? I think one cannot do it but is there some result about it?