A question on isolated points

Question

A scattered space is a space for which every not empty subset has an isolated point.

How to show it is equivalently for $T_1$ spaces, every not empty closed subset has an isolated point?

Thanks advance!

Answer 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 $A$.