Using the notation of Int, Cl and $\partial$ for interior, closure and boundary respectively, I'm trying to prove that Int$(\partial A) = \emptyset $, with A a closed set. However, the material that I find on the internet in this regard uses as a definition of boundary "The set of points $x \in X$ such that every neighborhood of $x$ contains at least one point of $S$ and at least one point not of $S$", while I'm using $\partial S = \text{Cl}(S) – \text{Int}(S)$.
I would like some hint on how to start the proof from the point of view of the definition shown above. Thanks!
Best Answer
It's quite easy to see that your definition of $\partial S = \text{Cl}(S) - \text{Int}(S)$ is equivalent to the one you saw: every neighbourhood of $x$ intersecting $S$ says $x \in \text{Cl}(S)$ and every neighbourhood of $x$ containing a point not in $S$ is the same as saying $x \notin \text{Int}(S)$.
Also consider $\partial A$ for $A$ closed, we write it as $ \text{Cl}(A) - \text{Int}(A) = A - \text{Int}(A)$, as $A$ is closed. So we subtracted from $A$ the maximal open subset of $A$ which means there cannot be any non-empty open subsets inside $A - \text{Int}(A) (\subseteq A)$, because that would contradict the maximality of $\text{Int}(A)$. Hence $\partial A$ has empty interior.