T2 Space is T1 Space

Theorem
Let $\left({S, \tau}\right)$ be a $T_2$ (Hausdorff) space.

Then $\left({S, \tau}\right)$ is also a $T_1$ (Fréchet) space.

Proof
From the definition of $T_2$ (Hausdorff) space:


 * $\forall x, y \in S: x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

As $U \cap V = \varnothing$ it follows from the definition of disjoint sets that:
 * $x \in U \implies x \notin V$
 * $y \in V \implies y \notin U$

So if $x \in U, y \in V$ then:


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

which is precisely the definition of a $T_1$ (Fréchet) space.