Dense-in-itself iff Subset of Derivative

Theorem

Let $T$ be a topological space.

Let $A \subseteq T$.

Then:

$A$ is dense-in-itself

if and only if:

$A \subseteq A'$

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.

$\blacksquare$