Empty Topological Space is Hausdorff

Theorem

Let $T = \struct {\O, \set \O}$ be the empty topological space.

Then $T$ is Hausdorff.

Proof

Recall the definition of Hausdorff Space:

$\struct {S, \tau}$ is a Hausdorff space or $T_2$ space if and only if:

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

That is:

for any two distinct elements $x, y \in S$ there exist disjoint open sets $U, V \in \tau$ containing $x$ and $y$ respectively.

This is vacuously true for the empty set.

$\blacksquare$