A question on isolated points
Solution 1:
$X$ need not be $T_1$: it’s true in general that if every non-empty closed subset of $X$ has an isolated point, then every non-empty subset of $X$ has an isolated point.
HINT: Suppose that every non-empty closed subset of $X$ has an isolated point, and let $A$ be any non-empty subset of $X$. Then $\operatorname{cl}A$ has an isolated point, say $x$. Now show that $x\in A$ and conclude that $x$ is an isolated point of $A$.