What are the prerequisites for topology?

There is no formal prerequisite except that if you haven't studied the topology of R and R^n and metric spaces first (as seen in elementary analysis courses), then the basic definitions of topology are going to be hard. In the sens that they will seem ferociously abstract and unmotivated.

 

Two good introductory books

A light hearted but mathematically sound overview
( watch out for the hairy ball theorem )
Surface Topology by P A Firby and C F Gardiner

A small more rigouous but easy to follow introduction

Introduction to Metric and Topological Spaces
By W A Sutherland

 

Point set topology is a really vast subject, but nobody needs to know all of it. There is a small core of concepts and results (almost all trvial from a mathematican's point of view) that any mathematically mature person is expected to know. The problem with Munkres if you're doing self study is that you won't know what the important concepts are because they are burried in a mass of less important ones. I consider Munkres more of a reference book in the sense that when on the rare occasions I encounter a point set topology concept I am unfamiliar with, I look it up in Munkres or some other point set topology book.

That said, I recommend the book 'Introduction to Topological Manifolds' by John M. Lee, chapters 2-3-4 and also Bredon's 'Topology and Geometry' chapter 0. Or course, you can also bring Munkres along but Lee should be your flashlight.

 

You'll want to learn point-set topology before algebraic topology. And learning real analysis will provide that. But even that aside, I'd still suggest learning some analysis before topology: it will be far easier to grasp homotopy and homeomorphisms once you have a handle on continuity (in the topological sense), compactness, images and inverses images of mappings of sets, and metric spaces. Calculus is so crude relative to topology, I don't think it provides any background for the subject at all.

 

You should definitely first study the topology of R^n and maybe in general metric spaces. Otherwise, it's going to be hard. You need to build up your mathematical muscles first. I suggest Rudin.

 

I don't think regular, undergraduate Calculus is enough to learn topology. You really need either:
1. A proofs class or
2. A logic class and knowledge of sets

 

The first chapter in the Topology by Munkres, covers pretty much everything you need to know on set theory and that will be essential in topology.

But i do agree that a proofs class is essential, or simply mathematical maturity for that matter. It is sad to see how some of my classmates struggle when it comes to proofs and the like. Not all of them of course, only the people who thought taking Topology as their first rigorous (strictly proof-based) course was a good idea.