Point is Isolated iff belongs to Set less Derivative

Theorem

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

Let $H \subseteq S$.

Let $x \in S$.

Then:

$x$ is an isolated point in $H$
$x \in H \setminus H'$

where

$H'$ denotes the derivative of $H$.

Proof

$x$ is an isolated point in $H$

$\iff$ $x \in H$ and $x$ is not an accumulation point of $H$ by Point is Isolated iff not Accumulation Point

$\iff$ $x \in H$ and $x \notin H'$ by definition of derivative

$\iff$ $x \in H \setminus H'$ by definition of set difference.

$\blacksquare$