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 $\overline{S^c} = S^{o^c}.$


1 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.


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