Equivalence of Definitions of Scattered Space

Definition

The following definitions of the concept of Scattered Space are equivalent:

Definition 1

A topological space $T = \struct {S, \tau}$ is scattered if and only if it contains no non-empty subset which is dense-in-itself.

That is, $T = \struct {S, \tau}$ is scattered if and only if every non-empty subset $H$ of $S$ contains at least one point which is isolated in $H$.

Definition 2

A topological space $T = \left({S, \tau}\right)$ is scattered if and only if it contains no non-empty closed set which is dense-in-itself.

That is, $T = \left({S, \tau}\right)$ is scattered if and only if every non-empty closed set $H$ of $S$ contains at least one point which is isolated in $H$.

Proof

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 .

$\Box$

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.

$\blacksquare$