In Scott topology, it is easy to show that principal ideals are closed sets (and their complements are open).
I suppose that in general case Scott topology is not necessarily generated by the complements of principal ideals (otherwise this could have been a more appealing definition). However, I can not yet come up with an example of a poset where I would see that Scott topology is not generated by the complements of principal ideals. Can anyone help, please?
I will also appreciate any insight into what purposes Scott's topology serves "better" than the topology generated by the complements of principal ideals.
One remarkable property of Scott topology is that continuous functions between posets with Scott topology preserve directed suprema (and in particular are monotone). However, to prove this, the following property $\mathbf{P}$ suffices:
The closure of every (upward) directed set is the (principal) ideal generated by its supremum.
(This property implies, in particular, that all closed sets are lower sets, and all principal ideals are closed.)
The topology generated by the complements of principal ideals is the weakest (coarsest) topology with property $\mathbf{P}$, while Scott topology is the strongest (finest) one with this property. How can I see that they are not the same?
Talking of continuous function, I see how to show that a function between two posets is Scott-continuous if (and only if) it preserves all directed suprema. Since a function continuous with respect to the topology generated by the complements of principal ideals will preserve all directed suprema, it will be also Scott-continuous. I would be rather surprised though if the converse was also true... Can anyone give me a counter-example to the converse?
To summarise, I am looking for two examples:
an example of a poset where Scott topology is not generated by the complements of principal ideals,
an example of a function between two posets that is continuous with repect to Scott topologies, but not continuous with respect to the topologies generated by the complements of principal ideals.