I am confused about definition of the boundary set. Definition: Given set $A$ and its complement $A^c$; Point $x$ belongs to the boundary of $A$ if every open ball centred at $x$ contains points of $A$ and $A^c$.
Now to my confusion: Taking as an example set $A\subset \mathbb{N}$ and its complement in $\mathbb{R}^1$. Set $A$ does not contain any limit points. Then for $\forall x\in A$ the only member of $A$ that is contained in all the balls centered at $x$, is $x$ itself. Now, $x\notin A^c$ but is a limit point in $A^c$, hence every ball centered at $x$ contains elements of $A^c$. Can I conclude that $x$ is in the boundary set of $A$, even though it is an isolated point in $A$? Also, say if the complement of $A$ is also in $\mathbb{N}$, will $x$ still be a boundary point of $A$?
Thanks a lot in advance! And I apologize for the clumsy language :-)
Leon