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

In summary, you should understand the basics of topology before doing differential geometry. If you want to look for a good material, I suggest chapters 2-4 of Lee, Topological Manifolds.
  • #1
JPMPhysics
3
0
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 :)
 
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  • #2
JPMPhysics said:
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.
 
  • #3
I disagree somewhat. My first exposure to differential geometry did not require any topology. The main prerequisites would be linear algebra and calculus.
 
  • #4
  • #5
JPMPhysics said:
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.
 
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1. What is Topology?

Topology is a branch of mathematics that deals with the study of shapes and spaces. It focuses on the properties of objects that remain unchanged when they are stretched, bent, or twisted, without being torn or glued together.

2. What are the most important definitions in Topology?

Some of the most important definitions in Topology include open and closed sets, continuity, compactness, and connectedness. These concepts are fundamental and essential to understanding the properties of topological spaces.

3. What are the most important results in Topology?

There are many important results in Topology, but some of the most significant ones include the Invariance of Domain theorem, the Brouwer Fixed Point theorem, and the Jordan Curve theorem. These results have applications in various fields, including physics, engineering, and computer science.

4. How is Topology used in real-world applications?

Topology has numerous practical applications in fields such as biology, chemistry, computer science, and economics. It is used to model and analyze complex systems, from biological networks to computer networks, and to understand the behavior of physical systems.

5. What are some common misconceptions about Topology?

One common misconception about Topology is that it is only concerned with the study of abstract and theoretical concepts. However, it has numerous practical applications, as mentioned earlier. Another misconception is that Topology is only relevant in pure mathematics, when in fact, it has significant applications in other disciplines as well.

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