Experiences using Brown's Topology and Groupoids in lieu of Munkres and Hatcher

Question

I recently became aware of the book Topology and Groupoids by Brown. After seeing a number of questions/answers here on Math.SE that referenced the book and had nice perspectives, I'm curious about its use as a somewhat introductory text for undergraduates.

In the past, I have always used Munkres and Hatcher as go-to textbooks. Has anyone else mentored/taught students using this book, and if so:

• How does it compare to these standard texts?
• What aspects does the book do well in, and what does it lack?
• What benefit is there, if any, in introducing a student early on to the perspective that Topology and Groupoids offers?

Answer 1

This is only a personal opinion.

What is obvious is that Topology and Groupoids doesn't have the same purposes as the Munkres and the Hatcher. As the title suggests, much importance is given to the algebraic structure of a groupoid, which is may not be the best way to tackle algebraic topology for the first time. But it is possible to read only the parts of the book which don't use the theory/language of groupoids. This is what I did and it was very helpful to me.

What I did really enjoy with this book is the part on function spaces, which is, I think, very well written. In general, the book is pretty much self-contained, in that all the non-basic topology you need is in it.

Of course, if you read the book except the parts on groupoids, there is way more content is the other books, but I would say that everything which is done is done very well, and in a style which is different than the others. I personally didn't read the Munkres very much so I can only really compare with the Hatcher.

What I like about this book is that it is has no shame to go directly into full generality, or almost. For example one of the first theorems about covering maps is that they are Hurewicz fibrations (more or less), which is a very general result which doesn't appear, I think, in the Hatcher (it does in the Spanier if I remember well). Despite the generality, I was never lost, as I could have been reading other books.

So, these are over all personal comments, but I hope this will answer partially your question. As a conclusion I would say that this book shouldn't be your only reference but it is a very good book to refer to whenever you have some uncertainty, and to read along another one.

Answer 2

The previous editions of 1968 and 1988 of Topology by Brown have been my favorite choices in both teaching and research. The present edition of 2006 contains many additional interesting topics. During the past several years, Semigroups have been extensively studied from the algebraic point of view [see the journal : Semigroup Forum]. Using the techniques developed in "Topology and Groupoid" by Brown, some results on Groupoid and semigroups can possibly be established in the setting of Topological Groupoids and Topological Semigroups. Prior to the knowledge of the present edition of the book, I have studied various topologies on Topological Algebras, Topological Modules and Modules of continuous homomorphism [published jointly in Demons. Math. 38(2005), Comtemp.Math. (2007, AMS) , JMAA 343(2008)].