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 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. Introduction There is no unanimously adopted localic analogue of the T1 axiom for topological spaces. Unordered (TU ) locales [10, 11], and subfit/conjunctive locales [17, 5] have been considered as candidates. However neither of these two notions is regarded as entirely satisfactory, primarily because both fail to coincide with the T1 property in the spatial case. For example there are Departments of Computer Science and Mathematics, University of Manchester. Department of Computer Science, University of Verona.