Learn Topology: Resources for Beginners

[dalcde]

I want to learn topology but I can't find any good resources (except for thoughtspacezero on youtube, which got a bit difficult to understand after a few videos). Can you suggest some materials?

Thanks in advance!

 

[dalcde]

I mean free resources or books you can find in normal public libraries (I don't have direct access to college or university libraries).

Thanks again!

 

[HallsofIvy]

Typically, "normal public libraries" do NOT carry books on topology!

However, you should be able to find websites that will help you.
The "MIT open courseware" site on "introduction to topology" at\
http://ocw.mit.edu/courses/mathematics/18-901-introduction-to-topology-fall-2004/
might be a good start.

 

[Monocles]

I studied topology on my own last summer by reading Topology by Munkres and doing all of the assignments on this page (including all of the problems that were not required):

http://people.math.gatech.edu/~etnyre/class/4431Fall09/index.html

I asked a professor for help whenever I got stuck for too long. It worked out great for me - I took a graduate level course that used Lee's Introduction to Smooth Manifolds the following semester and got an A. So, that is my anecdote on teaching oneself topology.

 

[lavinia]

In my opinion, basic topology should be learned through Complex Analysis..This approach supplies a context for topological ideas showing how they arise naturally from problems of understanding holomorphic functions. The ideas generalize easily to algebraic topology, differential topology, and the topology of metric spaces.

 

[dslowik]

http://www.math.uiowa.edu/~jsimon/COURSES/M132Fall07/?C=M;O=A

 

[alice chen]

Hi, I am a newbie in topology :D
I wanted to know if there are 2 homeomorphic manifold, can we make sure its diffeomorphic?
If there are 2 homeomorphic manifold but it isn't diffeomorphic, Can you give me some example??...
I am really sorry if my English is bad, I am Vietnamese :D

 

[micromass]

Hi, I am a newbie in topology :D
I wanted to know if there are 2 homeomorphic manifold, can we make sure its diffeomorphic?
If there are 2 homeomorphic manifold but it isn't diffeomorphic, Can you give me some example??...
I am really sorry if my English is bad, I am Vietnamese :D

What you are asking is very difficult. In turns out that in dimensions 1,2 and 3, all homeomorphic smooth manifolds are in fact diffeomorphic! Thus any example of homeomorphic, non-diffeomorphic manifolds will be an abstract example in a dimension higher than 3!

One such manifolds are the so-called exotic spheres. These are very cool manifolds which are homeomorphic with the unit sphere in $\mathbb{R}^n,n>3$, but not diffeomorphic. As you can guess, the construction and the study of exotic spheres is very difficult and it is still an active part of research! See http://en.wikipedia.org/wiki/Exotic_sphere

Other examples are Exotic $\mathbb{R}^4$'s, these are manifolds homeomorphic to $\mathbb{R}^4$ but not diffeomorphic. http://en.wikipedia.org/wiki/Exotic_R4

 

[dslowik]

http://en.wikipedia.org/wiki/Diffeomorphism
has a good discussion of this under the section:
Homeomorphism and diffeomorphism.

Here is a homeomorphism that is not a diffeomorphism:
f(x) = x for x on (0,1)
f(x) = 3x - 2 for x on [1,2)
is a homeomorphism of (0,2) with (0,4), but not a diffeomorphism (not differentiable at x=1).
But (0,1) is diffeomorphic with (0,4) via g(x) = 4x.

Is your question geometrically equivalent to: Can all the kinks be removed from a (topological) manifold?
(I say 'topological manifold' but I guess differentiable manifolds need not be second countable or Hausdorff?..)

 

[phys_student1]

I studied topology on my own last summer by reading Topology by Munkres and doing all of the assignments on this page (including all of the problems that were not required):

http://people.math.gatech.edu/~etnyre/class/4431Fall09/index.html

I asked a professor for help whenever I got stuck for too long. It worked out great for me - I took a graduate level course that used Lee's Introduction to Smooth Manifolds the following semester and got an A. So, that is my anecdote on teaching oneself topology.

Can I do the same without a background in neither Real analysis nor Complex analysis ?

(My Math Background includes Calculus I,II & II, Linear algebra, diff. eqns and probability)

 

[phys_student1]

In my opinion, basic topology should be learned through Complex Analysis..This approach supplies a context for topological ideas showing how they arise naturally from problems of understanding holomorphic functions. The ideas generalize easily to algebraic topology, differential topology, and the topology of metric spaces.

So the prerequisite is only complex analysis ?

What about Real analysis ?

 

[micromass]

You don't really need any prerequisites for topology. However, the more analysis courses you have taken, the better. I had my first topology class with real analysis already behind me, and it made life much easier. Familiarity with concepts such as metric spaces, open and closed sets, compactness can make topology much more intuitive.

So, I suggest to take real analysis first and then take topology. You don't need to do it this way (many people do it the other way), but I feel this is easiest...

 

[phys_student1]

You don't really need any prerequisites for topology. However, the more analysis courses you have taken, the better. I had my first topology class with real analysis already behind me, and it made life much easier. Familiarity with concepts such as metric spaces, open and closed sets, compactness can make topology much more intuitive.

So, I suggest to take real analysis first and then take topology. You don't need to do it this way (many people do it the other way), but I feel this is easiest...

Thank you very much, but I have seen many suggesting taking topology though complex
analysis, not real analysis, any ideas?