T1 Space is T0 Space

Theorem
Let $\struct {S, \tau}$ be a Fréchet ($T_1$) space.

Then $\struct {S, \tau}$ is also a Kolmogorov ($T_0$) space.

Proof
Let $\struct {S, \tau}$ be a $T_1$ space.

Let $x, y \in S: x \ne y$.

From the definition of $T_1$ space:
 * Both
 * $\exists U \in \tau: x \in U, y \notin U$
 * and:
 * $\exists V \in \tau: y \in V, x \notin V$

From the Rule of Simplification:


 * $\exists U \in \tau: x \in U, y \notin U$

From the Rule of Addition:


 * Either


 * $\exists U \in \tau: x \in U, y \notin U$
 * or:
 * $\exists V \in \tau: y \in V, x \notin V$

which is precisely the definition of a Kolmogorov ($T_0$) space.