topology

[ydoaPs]

tommarrow, i am going to order some books and i need some information:

i already have a background in differentiation and integration. what do i need to know before topology? is differential geometry before or after or the same as topology? any reccomendations on books would be nice as well.

[Dave]

Topologies are rather a different beast to anything you've touched there. Differentiation, integration and calculus etc is all very well and good, but you need to take it to the "next level", so to speak. Topological spaces attempt to basically extend the ideas of continuity, sequences and things like this to more abstract spaces than the reals. Along the way, you can touch on things like connected-ness, compactness, completeness and the like.

 

If I were you, I would strongly recommend reading up on analysis of real-valued functions, and study a lot of things such as continuity, etc. You might also find it useful to concentrate on analysis of functions such as [imath]f:\R^n \to \R^m[/imath], since a lot of the topics have a certain amount of overlap. When you've done this, you can take a look at metric spaces, and then wean yourself onto topologies.

 

It's by no means a simple option to take. Topologies are very abstract and can be quite hard to visualise. If you want some examples, I can give some

[ydoaPs]

i would like some examples. it is probably a good idea to see what i am getting myself into. could you also give a list of things beyond calculus that i should check out before i start?

[Tom Mattson]

I'm not a math expert, but I'll tell you what little I do know.

 

Knowledge of calculus will be about as helpful to you as knowledge of French art during the Rennaisance. What you will want to be grounded in is set theory. Also, geometry and topology are not the same thing. Both subjects can deal with the same objects (lines, circles, etc..) but they ask and answer different questions. Loosely speaking geometry deals with shape and topology deals with structure.

 

As for a book, the Schaum's outline called General Topology is pretty friendly. You can also dig around the following Link Directory, which I maintain. I know there are some free lecture notes there.

[matt grime]

Topology can be independent of Analysis in the sense you are thinking of on first reading. Topology (pointset) is in essence the algebraist's way of doing analysis. All you need from analysis is the idea that the real numbers have a distance notion that makes them a metric topological space.

[ydoaPs]

i found a book called Topology Essentials http://bordersstores.com/search/title_detail.jsp?id=3816165&srchTerms=topology+essentials&mediaType=1&srchType=Keyword. i am going to order that and Schuam's Outline of General Topology http://bordersstores.com/search/title_detail.jsp?id=2303185&srchTerms=topology+general&mediaType=1&srchType=Keyword. the first book looks like it should help me with all of the stuff that Dave and Tom said i should know first.

 

edit: I am not ordering them until monday, because i have 7 on 7 at Greensburg today and the linemen challenge.