Separation in Spaces

Abstract

We will start by recalling some standard separation axioms in topological spaces X and discussing how to cope with them without points. In the point-free setting we think of classical spaces in terms of the complete lattices Ω(X) of their open sets, and of general spaces as of complete lattices of similar nature. Hence, the fact that the classical separation is expressed by statements in which individual points (and non-open subsets) play a prominent role seems to be a principal obstacle for extending them to the more general context. But the situation is much more favourable than what one may expect. First of all, some of the separation conditions can be replaced by obviously equivalent statements using the calculus of the lattice only: such is an absolutely straightforward reformulation of normality, very easy (and classically transparent) reformulation of regularity, and a translation of complete regularity which needs some more explanation (but this explanation concerns the role of real functions which calls for explanation in the classical situation as well). This will be presented already in this chapter. Then there are reformulations and replacements that are more involved (in particular the Hausdorff axiom that will come in variants with nontrivial relations). And then there are specific point-free separation conditions that are of particular interest: some of them akin to classical ones but not quite equivalent, some of them naturally arising from lattice theoretic requirements. They will be discussed and analysed in the individual chapters below (to be more exact: regularity, complete regularity and normality will also have their individual chapters; what we have in mind is that their reformulations can be presented right away while the others will be postponed).