On the T1 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 ofa 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. (C) 2009 Elsevier B.V. All rights reserved.