# Scott Topological Lattice is T0 Space

## Theorem

Let $T = \left({S, \preceq, \tau}\right)$ be a complete topological lattice with Scott topology.

Then $T$ is a $T_0$ space.

## Proof

Let $x, y \in S$ such that

$x \ne y$
$\left\{ {x}\right\}^- = x^\preceq$ and $\left\{ {y}\right\}^- = y^\preceq$
$\left\{ {x}\right\}^- \ne \left\{ {y}\right\}^-$
$T$ is $T_0$ space.

$\blacksquare$