Point is Isolated iff not Accumulation Point

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

Let $x \in H$.

Then:
 * $x$ is an isolated point in $H$


 * $x$ is not an accumulation point of $H$
 * $x$ is not an accumulation point of $H$

Sufficient Condition
Let $x \in H$ be an isolated point in $H$.

Then by definition of isolated point:
 * $\exists U \in \tau: H \cap U = \set x$

That is, by definition of uniqueness:
 * $\lnot \forall U \in \tau: \paren {x \in U \implies \exists y \in S: \paren {y \in H \cap U \land x \ne y} }$

Hence by Characterization of Derivative by Open Sets:
 * $x \notin A'$

where $A'$ denotes the derivative of $A$.

Thus by definition of derivative:
 * $x$ is not an accumulation point of $H$.

Necessary Condition
Let $x \in H$ not be an accumulation point of $H$.

Thus by definition of derivative:
 * $x \notin A'$

Hence:

Thus by definition of isolated point:
 * $x$ is an isolated point in $H$.