# Should I study metric spaces topology before general topology?

#### Fantini

MHB Math Helper
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.

#### dwsmith

##### Well-known member
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.

#### Fantini

MHB Math Helper
In that case what would you recommend for a self-study?

#### dwsmith

##### Well-known member
In that case what would you recommend for a self-study?
Rudin or Royden.

#### Fantini

MHB Math Helper
By Rudin you mean the "Big" one, "Real and Complex Analysis"?

#### Ackbach

##### Indicium Physicus
Staff member
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.

#### dwsmith

##### Well-known member
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.

#### Fantini

MHB Math Helper
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.

#### PaulRS

##### Member
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!

#### mvCristi

##### New member
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. )

I vote for "General topology" if you like an abstract approach. More abstract, less details, more clear and easy.

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