Are these assertions about boundary, open, and closed sets correct?

Question

Are the assertions below about boundary, closed, and open sets correct? They help visualize several of the key concepts and are correspond more closely with intuition (at least my intuition).

$\partial S$ is the set of points $x$ such that for any $\varepsilon > 0$, there exists $s \in S$ and $r \notin S$ such that $x \in B_\varepsilon(s) \cap B_\varepsilon(r)$. That is, $x$ is arbitrarily close to (or in) both $S$ and its complement. This makes clear that $\partial S = \partial(S^c)$, and matches the intuitive sense of "boundary."

(This definition works for any metric space. I'm not sure how to extend it to a general topological space.)

A closed set is a set that includes its boundary. An open set is a set that's intersection with its boundary is empty. This makes clear that a clopen set is precisely a set with an empty boundary, and matches the intuitive sense of "open" and "closed".

The closure of a set is the union of the set and its boundary. The interior of a set is the set with its boundary removed. This makes clear that $\overline S^c = S^{c^o}$ and $\overline{S^c} = S^{o^c}.$

Answer 1

your definition of $\partial S$ is right though it is not the "standard "definition(of course it is equivalent to it):$$x\in \partial S \leftrightarrow \forall \epsilon >0,B(x,\epsilon)\cap S \neq \emptyset \land B(x,\epsilon)\cap S^{c} \neq \emptyset $$,the the definition stays the "same" (replacing open balls which are basic nbds with open subsets or a system of open nbds) in the case of topological space: $$ x\in \partial S \leftrightarrow \forall U\in V_{x},U\cap S \neq \emptyset \land U\cap S^{c} \neq \emptyset $$ where $V_{x}$ denote the set of nbds of $x$.the rest of characterizations are also right.