Equivalence of Definitions of Scattered Space

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Definition 1 implies Definition 2
Let $T$ be defined as in definition 1.

That is, $T$ contains no non-empty subset which is dense-in-itself.

Let $H \subseteq T$.

Then whether $H$ is closed or not, it is not dense-in-itself.

In particular, if $H$ is closed, then it is not dense-in-itself.

Hence $T$ satisfies definition 2.

Definition 2 implies Definition 1
Now let $T$ be defined as in definition 2.

Let $H \subseteq T$.

Then $H^-$ is closed, where $H^-$ denotes the closure of $H$.

From Topological Closure is Closed, $H^-$ is closed.

Because $T$ satisfies definition $(2)$, $H^-$ has an isolated point by definition.

From Isolated Point of Closure of Subset is Isolated Point of Subset, we conclude that $H$ also has an isolated point.

Hence $T$ satisfies definition 2.