T1 Space is T1/2 Space

Theorem
Let $T$ be a $T_1$ topological space.

Then $T$ is $T_{\frac 1 2}$ space.

Proof
By Closure of Derivative is Derivative in T1 Space:
 * $\forall A \subseteq T: \left({A'}\right)^- = A'$

where
 * $A'$ denotes the derivative of $A$
 * $\left({A'}\right)^-$ denotes the closure of $A'$

Then by Topological Closure is Closed:
 * $\forall A \subseteq T: A'$ is closed

Thus by definition:
 * $T$ is $T_{\frac 1 2}$ space