Lawson topology

Remarks

Example: on $P = [0, 1]$ with the usual order, the Scott open sets are $\emptyset$, $[0, 1]$ and the intervals $(a, 1]$ with $0 \le a < 1$. The lower open sets are generated by the intervals $P \setminus {\uparrow} b = [0, b)$. Finite intersections of these sets give all intervals $(a, b)$, $[0, b)$ and $(a, 1]$, so the Lawson topology is the usual topology of $[0, 1]$.

Non-example: the Scott topology of $[0, 1]$ is not its Lawson topology. The set $[0, 1/2)$ is Lawson open, but it is not an upper set, so it is not Scott open.

The Lawson topology of a continuous lattice is compact and Hausdorff. Some authors define the Lawson topology only for directed-complete partial orders or for continuous posets, where it is best behaved.