# T1 Space is T0 Space

## Theorem

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

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

## Proof

Let $\left({S, \tau}\right)$ 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:

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

$\exists U \in \tau: x \in U, y \notin U$
$\exists V \in \tau: y \in V, x \notin V$
which is precisely the definition of a Kolmogorov ($T_0$) space.
$\blacksquare$