In the space of integers with usual topology, is it true that...
In real space and space of rationals...
These are my first thoughts when I read the definition of isolated points and derived sets. Just wanted to verify if they are correct
Your first line on integers is correct.
The set of integers $\mathbb Z$ is usually equipped with the discrete topology, which means that every subset of $\mathbb Z$ is an open set. Singletons are open sets and the statement that $\{x\}$ is an open set is exactly the same as the statement that $x$ is an isolated point of the whole space $\mathbb Z$.
Secondly if $x$ is an isolated point of the whole space, then automatically it is an isolated point of every set $S$ with $x\in S$.
Your second line on reals and rationals is not correct.
If $x\in X$ where $X=\mathbb Q$ or $X=\mathbb R$ then (if we are working with the usual topology on $X$) then singletons are evidently not open sets, or equivalently: no $x\in X$ is an isolated point of the whole space $X$.
Secondly (and this is what I mark as your incorrectness) if $S\subset X$ and $x\in S$ then quite well it can happen that $x$ is an isolated point of subset $S$. This is the case if and only if there is an open set $U$ such that $U\cap S=\{x\}$. Taking e.g. $S=\{0\}\cup\{z\in X\mid z>1\}$ we come to the conclusion that $0$ is an isolated point of $S$, since $U\cap S=\{0\}$ for open set $U=\{z\in X\mid z<1\}$.
