Should I study metric spaces topology before general topology?

In summary: If you have the knowledge for it, yes. If not, start little and then go big. I'll look for Crossley and Kirkwood books you mentioned, Ackbach.
  • #1
Fantini
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Hello everyone. I want to study topology ahead of time (it begins next semester only) and I have two options: I could go straight for general topology (among the books I searched I found Munkres to be the one I felt most comfortable with) or go for a thorough study of metric spaces (in which case I'm open for book suggestions as well). I would like to know what are your recommendations in this case and what advice that you think could be useful.

Thanks for all.:)
 
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  • #2
Fantini said:
Hello everyone. I want to study topology ahead of time (it begins next semester only) and I have two options: I could go straight for general topology (among the books I searched I found Munkres to be the one I felt most comfortable with) or go for a thorough study of metric spaces (in which case I'm open for book suggestions as well). I would like to know what are your recommendations in this case and what advice that you think could be useful.

Thanks for all.:)

Metric space topology I would say is just real analysis.
 
  • #3
In that case what would you recommend for a self-study?
 
  • #4
Fantini said:
In that case what would you recommend for a self-study?

Rudin or Royden.
 
  • #5
By Rudin you mean the "Big" one, "Real and Complex Analysis"?
 
  • #6
Real Analysis is, in my mind, too important a subject to be left to only one book. I'd start with Kirkwood's Introduction, then do Baby Rudin. I'd probably take a break, do some other math like complex analysis, and then come back and do Royden and Papa Rudin.

As for topology, have you considered Crossley's Essential Toplogy? I've found what I've read so far of that to be eminently readable, and a good intro.

Note: Baby Rudin = Principles, Papa Rudin = Real and Complex. Grandpapa Rudin = Functional, I suppose.
 
  • #7
Fantini said:
By Rudin you mean the "Big" one, "Real and Complex Analysis"?

If you have the knowledge for it, yes. If not, start little and then go big.
 
  • #8
I'll look for Crossley and Kirkwood books you mentioned, Ackbach. I believe it's best if I mention some of my math background:

I'm done with all calculus 1-3 (single, multivariable and differential equations); done linear algebra and advanced linear algebra (PhD level); did Analysis I about a year ago but it was horrible, I learned almost nothing. The teacher focus' was more on calculus than proofs and intuition, which resulted in a pass without the appropriate maturity and knowledge developed.

As of now, I'm starting Analysis II, Analysis in \( \mathbb{R}^n \) (or on Manifolds) and Groups and Representations. As for books used, in the bibliography for the first there's "Methods of Real Analysis" by Goldberg, the teacher in \( \mathbb{R}^n \) is using "Calculus on Manifolds" by Spivak, but has said he'll use at times "A Comprehensive Introduction to Differential Geometry, volume 1" for a few things (I've also decided on taking Munkres' "Analysis on Manifolds" as a reference, studied differentiation using it before classes began and I enjoyed it), last but not least there were no recommendations for Groups, it was said that any book containing the basic ideas of groups would suffice, and I chose Rotman's "Introduction to the Theory of Groups" as my guide.

The idea of studying topology/metric spaces would be in parallel with those three.
 
  • #9
The book on general topology that I liked the most is A taste of topology , it is pure gold!

Chapter 1 : Set Theory (Axiom of Choice / Zorn's Lemma, Cantor-Bernstein , Countability, etc.)
Chapter 2 : Metric Spaces
Chapter 3 : Topological Spaces
Chapter 4 : Systems of continuous functions ( Urysohn's Lemma, etc. )
Chapter 5 : Basic Algebraic Topology

And all of this in roughly 200 pages, plus beautiful exposition! :)
 
  • #10
All I can say, I remember well what general topology means. However, I also remember when topology was mixed with Banach spaces and Hausdorff spaces, just to approach Weirstrass calculus from general and more particular point of views. It was as obscure as it sounds, but it was correct, so I cannot object. I remember it was mixed, but I do not want to remember anything about it. (I still have nightmares about the exam. :p)

I vote for "General topology" if you like an abstract approach. More abstract, less details, more clear and easy.
 
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  • #11
The book "A Taste in Topology" has interested me, primarily because it contains most topics I need for related areas. I hope it is as good as you speak. Thanks for the mention! As for the general topology, I think I won't have the time now. Besides, I'm not sure it would entirely help me in a very short time span, meaning I can postpone without problems for a while.
 

FAQ: Should I study metric spaces topology before general topology?

Should I study metric spaces topology before general topology?

It is generally recommended to study metric spaces topology before general topology, as it provides a foundation for understanding the concepts and techniques used in general topology. Metric spaces topology focuses on the study of topological spaces with a distance function, providing a more concrete and intuitive approach to topology. Understanding metric spaces topology can also make it easier to grasp the more abstract concepts in general topology.

What is the difference between metric spaces topology and general topology?

Metric spaces topology is a specialized branch of general topology that focuses on the study of topological spaces equipped with a distance function, while general topology is a broader branch that deals with the properties and structures of topological spaces in general. Metric spaces topology is more concrete and intuitive, while general topology is more abstract and theoretical.

Can I skip studying metric spaces topology and go straight to general topology?

While it is possible to skip studying metric spaces topology and go straight to general topology, it is not recommended. Understanding metric spaces topology can make it easier to grasp the concepts and techniques used in general topology. It also provides a foundation for understanding topological structures and properties in a more concrete and intuitive manner.

What are some real-world applications of metric spaces topology?

Metric spaces topology has various real-world applications in fields such as physics, engineering, and computer science. It is used to study physical spaces and their properties, such as the distance between objects in a 3-dimensional space. It is also used in optimization problems and data analysis, such as clustering algorithms in machine learning.

Are there any prerequisites for studying metric spaces topology?

A basic understanding of mathematical concepts such as sets, functions, and basic analysis is helpful for studying metric spaces topology. Some familiarity with linear algebra and real analysis may also be beneficial. However, there are no strict prerequisites, and anyone with a strong foundation in mathematics can study metric spaces topology.

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