Is there an introduction to uniform structures besides Bourbaki?

Question

Chapter 2 of Bourbaki's General Topology is the only introduction to uniform structures on topological spaces of which I am aware.

However, for me it is very time-consuming to extract any understanding at all from Bourbaki's writing. Because of this I would be very grateful for an introduction to the topic which is at a more elementary level, even though I understand that Bourbaki is the definitive reference.

Is there an introduction to uniform structures besides Bourbaki?

Answer 1

If you read German (as you can from the comments), Herrlich's 2 book series Topologie I (Topologische Räume), Topologie II (Uniforme Räume) are quite accessible, and have lots of exercises. You can use the first book to get acquainted with his terminology (which is sometimes non-standard) and the uniform spaces part start completely from scratch, connecting the notions to topological ones from book 1.

Also, James' book on Topologies and Uniformities (Springer Undergraduate series) is also pretty easy-going.

Warren Page's book on uniform structures is very oriented towards functional analysis and quite formal in its treatment; it covers topological groups and vector spaces quite early on, as well as Haar measure and function spaces. It's more of a reference book than James and Herrlich's (which are self study-able books)

Engelking (General Topology) has a chapter on uniform spaces as well. But that is pretty basic. Just to get the flavour, really, and see the connection to topologies. This from browsing my shelves.

Answer 2

Metric notions (such as well-chainedness, completeness, precompactness, and uniform continuity, for instance) can be best clarified and treated in relator spaces (X, Y)(Cal R) consisting of two sets X, Y and a family Cal R of relations on X to Y.

There, it turns out that connectedness and compactness are particular cases of well-chainedness and precompactness. Moreover, convergent and Cauchy are actually equivalent notions. Furthermore fat and dense sets are usually more important than the open and closed ones. And instead of continuities of functions it is better to define and investigate those of relations and relators.

Unfortunately, the publications of our papers on relators are frequently prevented by the leading topologists acting in editorial boards of various journals. And we cannot even publish in the mathematical journals of Debrecen and Budapest.

Several authors study generalized topologies instead of generalized uniformities. Moreover, relators are also completely ignored by the editors of the Mathematical Subject Classification.