Equivalence of Definitions of Scattered Space

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

Consider the definitions for $T$ to be defined as a scattered space:
 * $(1): \quad T$ is scattered iff it contains no non-empty subset which is dense-in-itself.
 * $(2): \quad T$ is scattered iff it contains no non-empty closed set which is dense-in-itself.

These two definitions are logically equivalent.

Proof
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 $(2)$.

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

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

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

Hence $T$ satisfies definition $(1)$.