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:

$\forall A \subseteq T: \paren {A'}^- = A'$

where

$A'$ denotes the derivative of $A$

$\paren {A'}^-$ 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

$\blacksquare$

Sources

• 1989: Ryszard Engelking: General Topology (revised and completed ed.)

• Mizar article TOPGEN_4:46