Any good problem book on General Topology

In summary, there are various problem books available for general topology, ranging from those with simple exercises to more complex and even open problems. It is recommended to have a background in real analysis before studying topology. Topology is also used in mathematical physics and can be helpful in understanding concepts such as closed timelike curves.
  • #1
huyichen
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I found it is hard for myself to follow the book on general topology by willard, since there are too many abstract definitions with too few examples to help me to establish these terms. I am wondering if there is any good problem book with sufficient problems that would help to make abstract concepts more concrete. Any suggestion will be appreciated.
 
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  • #2
I don't know about other books, but with the majority of examples, you're really just finding ways to describe plain subsets R^n. Want a topological space? R^n. Want a compact space? A closed sphere, box, or any other shape with finite volume. Want a homeomorphism? Morph a sphere into a box. Connected spaces? R^n. Disconnected spaces? A disjoint box and sphere. Product spaces? The pairs of points on a disjoint sphere and box. Quotient spaces? A sphere where opposite points are considered equal.

For the most part, books like that one will take the same darned examples and use them over and over again. There are more exotic spaces with less intuitive properties (such as L^infinity), but even those examples tend to get reused in the book.
 
  • #3
the thing is that some times it is hard to follow the book, since the problem provided as exercises have no answer, so it is hard to judge by myself whether I have the correct answer or not, so at least a solution manual for any book on general topology should be helpful, so is there any book together with solution manual available?
 
  • #4
These two are reasonable for problems:

Topology Problem Solver (Problem Solvers) (Paperback) by REA

and of course,

Schaum's Outline of General Topology (Paperback)

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For insight:
As far as textbooks, my favorite is the trilogy by John M Lee starting with:
Introduction to Topological Manifolds. It's beginning graduate level but is par excellence on motivational insight. A great undergraduate text for insight is:
Topology of Surfaces (Undergraduate Texts in Mathematics) by L.Christine Kinsey
and there are many Dover entries. For example:
Topology: An Introduction to the Point-Set and Algebraic Areas by Donald W. Kahn
is very approachable (more so than Willard).

Finally the two MIT classics are the undergrad texts by Munkres and Singer & Thorpe.

All of the above assumes a background in real analysis "There's a delta for every epsilon" right? If that's new to you, Real Analysis by Frank Morgan is a great introduction. Don't try to push brute force through point set topology without knowledge of real analysis.
 
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  • #5
Thanks, well, I have finished both real analysis and complex analysis, so the basic ideas like compactness or connectness or continuity are clear to me. The problem books you mentioned are also the only book i found by myself.
 
  • #6
I like Topology by Janich. It gets the intuition across, but it doesn't have any exercises.
 
  • #8
huyichen said:
I found it is hard for myself to follow the book on general topology by willard, since there are too many abstract definitions with too few examples to help me to establish these terms. I am wondering if there is any good problem book with sufficient problems that would help to make abstract concepts more concrete. Any suggestion will be appreciated.

why do you want to study topology? I think the general conceptions is useless in physics.if you don't use it ,you can't master it.it is just waste of time for physicists to study it. you 'd better abandon it.
'
 
  • #9
bobydbcn said:
why do you want to study topology? I think the general conceptions is useless in physics.if you don't use it ,you can't master it.it is just waste of time for physicists to study it. you 'd better abandon it.
'

Are you serious?

Firstly, this a topology forum.

Secondly, huyichen, the original poster in this thread has not mentioned physics, and might have no interest in physics. For example, for all we know, huyichen could be a pure mathematician or a student of pure mathematics who has no interest in physics.

Thirdly, topology is used in various areas of mathematical physics, e.g. the global methods used in general relativity. I think it would be difficult to read the proofs in Hawking and Ellis with understanding without some knowledge of topology.

For example, it's easy to show using a topological argument,

https://www.physicsforums.com/showthread.php?p=1254758#post1254758,

that any compact spacetime must have closed timelike curves (time travel).
 
  • #10
huyichen said:
I found it is hard for myself to follow the book on general topology by willard, since there are too many abstract definitions with too few examples to help me to establish these terms. I am wondering if there is any good problem book with sufficient problems that would help to make abstract concepts more concrete. Any suggestion will be appreciated.

personally I advise you to learn general topology as you need it in other areas of mathematics. usually a first course in complex analysis will give you a strong start.
 
  • #11
George Jones said:
Are you serious?

Firstly, this a topology forum.

Secondly, huyichen, the original poster in this thread has not mentioned physics, and might have no interest in physics. For example, for all we know, huyichen could be a pure mathematician or a student of pure mathematics who has no interest in physics.

Thirdly, topology is used in various areas of mathematical physics, e.g. the global methods used in general relativity. I think it would be difficult to read the proofs in Hawking and Ellis with understanding without some knowledge of topology.

For example, it's easy to show using a topological argument,

https://www.physicsforums.com/showthread.php?p=1254758#post1254758,

that any compact spacetime must have closed timelike curves (time travel).

Thanks a lot!your criticism opened my vision. Now I just read books about differential geometry in physics.I am just a student.
 

FAQ: Any good problem book on General Topology

1. What is General Topology?

General Topology is a branch of mathematics that studies the properties of topological spaces, which are mathematical structures that generalize the concept of a metric space. It deals with the concepts of continuity, convergence, compactness, and connectedness.

2. Why is it important to have a good problem book on General Topology?

A good problem book on General Topology is important because it allows students and researchers to practice and deepen their understanding of the subject. It also helps them develop problem-solving skills and prepares them for more advanced topics in topology and other branches of mathematics.

3. How do I choose a good problem book on General Topology?

When choosing a problem book on General Topology, it is important to consider the level of difficulty, the variety of problems, and the clarity of explanations. Look for books that have a good balance of theory and applications, and make sure to read reviews and ask for recommendations from other students or experts in the field.

4. Can a problem book on General Topology be used for self-study?

Yes, a good problem book on General Topology can be used for self-study. However, it is recommended to have a solid foundation in basic topology and some knowledge of mathematical proofs before attempting to study from a problem book on General Topology.

5. Are there any online resources for problem solving in General Topology?

Yes, there are several online resources for problem solving in General Topology, such as online forums, websites, and video lectures. These resources can provide additional practice problems and solutions, as well as explanations and discussions on various topics in General Topology.

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