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

In this note a T 1 formal space ( T 1 set-generated locale) is a formal space whose points are closed as subspaces. Any regular formal space is T 1 . We introduce the more general notion of a T 1 ∗ formal space, and prove that the class of points of a weakly set-presentable T 1 ∗ formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulate separation properties T i ∗ for constructive topological spaces (ct-spaces), strengthening separation properties discussed elsewhere. Finally we relate the T i ∗ properties for ct-spaces with corresponding properties of formal spaces.