Separation in Point-Free Topology

Topology, classically, is commonly considered in flavors, two of which are point-set topology and algebraic topology. These two flavors are so distinct, each with very different toolsets and driving problems, that it would only be a minor exaggeration to say that the only thing they have in common is the underlying notion of what a topology is. 

 

Point-set topology views a topology as extra structure upon a set of points. Its motivating problems are, increasingly, set-theoretic in nature. The point-set topological notion of sameness is homeomorphism, which is necessarily a bijective function of the underlying sets.  Algebraic topology has a very different notion of sameness, namely homotopy equivalence. Two spaces can have radically different underlying sets while still being homotopically equivalent. In this respect algebraic topology cares much less about the points of a topological space. 

 

This level of interest in the points of a space dictated much of the development of the subjects. In modern terminology we can say that point-set topology is primarily interested in the category Top of topological spaces, and categories built directly out of it. Algebraic topology, on the other hand, is interested in the infinity category of simplicial sets. There is, however, a question that can, and was, asked very early on: can we get rid of the points in a topological space altogether? 

 

The answer is a very straightforward ‘yes’, as follows. A topological space is a set together with a collection of subsets that are termed ‘open’. The axioms of a topology state that this collection of open sets is stable under finite intersections and arbitrary unions. Viewed this way, the set of points is merely there to allow us to speak of subsets of it. Recognizing that the structure of the open subsets is what we really care about paves the way to getting rid of the points. This is done by defining a frame to be a complete join lattice in which meets distribute over joins. Then the collection of open sets in any topological space is a frame, but not every frame is the collection of open sets of a topological space. In a frame, there are no points. 

 

Point-free topology is the study of frames, from a geometric perspective. Without a doubt, point-set topology received more attention than its point-free sister. Superficially, one might expect there to be little difference between the two subjects. Perhaps there might even be a (more or less) mechanical process of converting theorems back and forth. This, however, is not the case. Famously, the point-free version of Tychonov’s theorem does not rely on the axiom of choice.

 

One place where one should expect some challenge when ‘translating’ from point-set topology to point-free topology is in the statement and meaning of the separation axioms. One needs to find a point-free way to speak of the separation of, well, points. This exercise proved very challenging, interesting, and fruitful. It is also the entry point of the book under review here. 

 

The book is written in a very engaging style. Its authors span between them a century of experience and much more than a handful of research results that shaped and advanced point-free topology. Their written words carry that weight very elegantly. 

 

It is very likely that anybody contemplating reading the book already knows some topology. In that case it is strongly advised to approach the book with an open mind and to expect a very different approach to arguing about topological notions. The benefits of doing so are strewn throughout the text and the first few chapters are a rather gentle introduction to the whole idea. The prerequisites for reading the book are rather limited, and can be found in the appendix. However, in my opinion, the prerequisites for enjoying the book include a sufficient level of comfort with the language of lattices from the perspective of category theory. Thus, if a reader is finding herself in deep waters and is seeking a lifeline in the appendix, it is probably a good idea to then spend significant time in the appendix, treating it as an opportunity to hone one’s lattice theoretic skills in general. 

 

I find the book to be an extremely valuable addition to the literature. Currently, there is an abundance of books on classical topology, but only a few that take the point-free approach. This situation is a reflection of history, not of difficulty or importance. The authors certainly kept that in mind. Chapter I immediately delves into the subject matter, but is written also as an introduction to the background concepts. Chapter II focuses of subfitness, a notion of separation that is truly unique to the point-free approach. The Hausdorff separation axiom, so simple in the classical case, occupies Chapter III, where the intricacies involved are beautifully narrated. Chapter IV presents the tangled web of interrelations between the separation concepts, illustrating clearly the difference compared to the classical setting. This chapter also marks the passage of the book to more advanced topics and classical results from frame theory. 

 

To conclude, the book is appealing to experts who will find a clear exposition of numerous results elegantly bundled into a coherent story. The book is also valuable to those immigrating from point-centric topology to the point-free realm. 

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Dr. Ittay Weiss did his PhD at Utrecht University, The Netherlands. He is interested in applied category theory and is a co-founder of Quantale.ai.