Separation Axioms

Abstract

In this chapter we will discuss the counterparts (or, almost counterparts) of the separation axioms Ti. Although the standardly used definitions seem to be heavily dependent on points, it is only partly so: for instance regularity and complete regularity can be perfectly reformulated in the point-free language. Instead of T1 we can consider a weaker version that has, however a very good interpretation in classical topology. What is hard to replace is the Hausdorff axiom; however, there are formally very similar conditions (not quite corresponding to the classical concept factually) that turn out to be very useful.