Dense-in-itself iff Subset of Derivative

Theorem
Let $T$ be a topological space.

Let $A \subseteq T$.

Then:
 * $A$ is dense-in-itself


 * $A \subseteq A'$
 * $A \subseteq A'$

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

Proof

 * $\qquad A$ is dense-in-itself


 * $\leadstoandfrom$ every $x \in A$ is not an isolated point in $A$ by definition of dense-in-itself


 * $\leadstoandfrom$ every $x \in A$ is an accumulation point of $A$ by Point is Isolated iff not Accumulation Point


 * $\leadstoandfrom$ every $x \in A$ is an element of $A'$ by definition of derivative


 * $\leadstoandfrom A \subseteq A'$ by definition of subset.