 
Summary: On the T1 Axiom and Other Separation
Properties in Constructive Pointfree and
Pointset Topology
Peter Aczel
Giovanni Curi
First draft: September 25, 2005; this version: December 27,
2008
Abstract
In this note a T1 formal space (T1 setgenerated locale) is a for
mal 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 setpresentable T
1 for
mal 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 (ctspaces), strength
ening separation properties discussed elsewhere. Finally we relate the
T
