# Supplementary book for topology

• Topology
I am taking a course in topology with Gamelin and Greene, Introduction to topology. I would like to have some supplement to extend and give more motivation and explanation. I am quite tired of the "theorem, proof, theorem, proof" pattern.

Thank you!

## Answers and Replies

Topology is meanwhile a wide range of rather different areas. You will find some (but not all) here:
https://www.ams.org/open-math-notes (search for topology)

I assume what you are looking for maybe this one:
https://www.ams.org/open-math-notes/omn-view-listing?listingId=110653

Thanks for the links.
I am looking for a text which, as Gamelin and Green, tends more towards analysis and its application than just general topology.

Thank you again.

fresh_42
Mentor
Thanks for the links.
I am looking for a text which, as Gamelin and Green, tends more towards analysis and its application than just general topology.

Thank you again.
In this case, you should look for measure theory and sigma-algebras.

mathwonk
Science Advisor
Homework Helper
2020 Award
to get some intuition for topology i was going to suggest to you to read some of hilbert and cohn vossen's geometry and the imagination, or courant's what is mathematics. but after what you said last i am puzzled as to what you want exactly. john kelley famously said he was originally wanting to title his General Topology, as "what every young analyst should know". i.e. although called general topology it is aimed exactly at analysis students. so i suggest you go to a university math library and sit in the topology section and see what book has what you want. topology is useful in analysis because it gives you easy methods that allow you to deduce at least qualitative results about analysis. the first example is the intermediate value theorem, which lets you prove that every real polynomial of odd degree has a real root without being able to find one exactly. i.e. topology tells you that if f(a) < 0 and f(b) >0 and f is continuous on [a,b] then there is a point c somewhere in (a,b) with f(c) = 0. (In fact if you weight the roots by their multiplicities you can prove there is an odd [weighted] number [or an infinite number] of roots.) In complex analysis topology lets you prove every polynomial f:C-->C on the complex plane has a complex root, since it extends to a map of the projective line to itself which is both open and closed, hence surjective, but only infinity goes to infinity, hence as a map C--->C it is surjective. when trying to solve the riemann roch problem of computing the number of independent meromorphic functions on a compact riemann surface with given pole divisor, one tweaks it by asking instead for the difference between that number and the number of independent holomorphic differentials with zeroes along that divisor. This difference turns out to be a topological invariant and hence can be calculated by degeneration to a special case. these techniques even can be jazzed up to prove such high powered theorems as the atiyah singer index theorem. so topology is the art of deforming your problem, and it is useful for computations after determining which aspects of your problem remain unchanged after deformation. you may have heard that it is possible to "evert a sphere" or turn a sphere in space inside out without introducing a kink. This is proved by showing the space of all immersions of the sphere is "connected" (a topological concept), hence there is a path of immersions joining the usual immersion to the antipodal one.

well on amazon, most people seem to recommend munkres:

https://www.amazon.com/dp/9332549532/?tag=pfamazon01-20&tag=pfamazon01-20

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