How can I find interior of $S$, exterior of $S$, Boundary of $S$, limit points of $S$ ,isolated points of $S$ and closure of $S$?
My try:
I found that $ Int(S)=\phi$ because if $m \in S$ there doesn't exist open set $U \in$ usual topolgy of $R$ such that $m \in U \subseteq S$.
But sadly I was not capable of finding the rest of them.
My motivation to problem: Yesterday at night I was searching about proofs of irrationality of $e$ and I found pdf file that give outline which is built as exercises for proof using topology...one exercise of them is:
Prove that $\alpha \in \mathbb{Q}$ iff $0$ is an isolated point of $\{a+b\alpha:a,b \in \mathbb{Z}\}$.
So for this reason I become curious about finding interior of S, exterior of S, Boundary of S, limit points of S ,isolated points of S and closure of S (note:I choose $\alpha=\sqrt{2}$ in $S$).
The pdf file is: https://web.williams.edu/Mathematics/lg5/Irrationale.pdf
This obviously doesn't answer my question.
– Mahmoud albahar Aug 19 '22 at 06:07I asked how to find them not just what are they equal to.
– Mahmoud albahar Aug 19 '22 at 06:08Where is the explanation of how to find boundary points or isolated points of $S$ in the question you suggested?
and the question is about someone discussing his arguments(that don't answer my question) and ask whether they are false or true .
And finally thank you for your try to help by suggesting similar question.
– Mahmoud albahar Aug 19 '22 at 06:18