On the T1 axiom and other separation properties in constructive point-free and point-set topology

In this note a T1 formal space (T1 set-generated locale) is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of a formal space, and prove that the class of points of a weakly set-presentable formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulate separation properties for constructive topological spaces (ct-spaces), strengthening separation properties discussed elsewhere. Finally we relate the properties for ct-spaces with corresponding properties of formal spaces.