A briefing of Topology's most important definitions and results?

[JPMPhysics]

I just need to know the basic ideas of topology, and the most important results, because I'll have differential geometry the next semester. Does anyone have a good material for this? Or you can just say what to search for and I'll search it. Thank you :)

 

[Mandelbroth]

I just need to know the basic ideas of topology, and the most important results, because I'll have differential geometry the next semester. Does anyone have a good material for this? Or you can just say what to search for and I'll search it. Thank you :)

You really should understand topology pretty well before you do differential geometry. I found Munkres to be a good start, if you want to look for it. I happen to know that there are a couple copies wandering the internet.

The basics:

A topology, which I shall call $\mathscr{T}$, on a space $S$ is a collection of sets contained in $S$ such that the following are satisfied:

1. $S$ and $\emptyset$ are in $\mathscr{T}$
2. The union of elements in any subcollection of $\mathscr{T}$ is in $\mathscr{T}$
3. The intersection of elements in any finite subcollection of $\mathscr{T}$ is in $\mathscr{T}$

Elements of the topology are called open sets. A pairing $(S,\mathscr{T})$ is called a topological space.

 

[mathman]

I disagree somewhat. My first exposure to differential geometry did not require any topology. The main prerequisites would be linear algebra and calculus.

 

[johnqwertyful]

Really, Topology is just an abstraction of Analysis. Topological spaces are just an extension of metric spaces, dropping the metric in favor of open sets.

http://en.wikipedia.org/wiki/Tychonoff's_theorem

Is important

 

[Jorriss]

I just need to know the basic ideas of topology, and the most important results, because I'll have differential geometry the next semester. Does anyone have a good material for this? Or you can just say what to search for and I'll search it. Thank you :)

Chapters 2-4 of Lee, Topological Manifolds might be useful.