Learning Topology: Problem Solving & Book Recommendations

[Avatrin]

Hi

I have to learn some general topology within the next two months. My experience with learning is that I learn better through problem solving; 'The Fundamental Theorem of Algebra' by Fine and Rosenberger helped me a lot when I was learning abstract algebra. So, I am looking for problems that are easy to understand but hard to solve unless you know some topology.

Also, the book I am using is Munkres' Topology. However, I hope somebody here can recommend a book that is better at providing motivation for the concepts I need to learn.

 

[micromass]

An awesome book that is really filled with motivation in Janich https://www.amazon.com/dp/0387908927/?tag=pfamazon01-20
It is one of the best topology books out there. But you absolutely need to supplement it with a different book since it is not that rigorous. Janich and Munkres together would work well.

 

[FactChecker]

Check Schaum's Outline Series "Theory and Problems of General Topology" by Seymour Lipschutz. All of the Shaum's outlines are full of examples and worked problems. General topology is motivated by the same concepts showing up and being repeated in many situations rather than being due to one large problem being solved.

 

[Avatrin]

I am looking through Janich's book, and I am not sure it is what I am looking for. I do not think Lipschutz' book is what I need either (it is just a collection of solved problems). So, I would like more suggestions

General topology is motivated by the same concepts showing up and being repeated in many situations rather than being due to one large problem being solved.

I need to clarify what I meant when I wrote that I want to see problems that are easier to solve with topology than without. I wasn't thinking about historical problems that lead to the creation of the field; Rather, I meant problems that I could work on to motivate myself to work faster. Also, to connect topology to things I already know (real analysis, abstract algebra, complex analysis, calculus etc).

 

[mathwonk]

connectedness in topology is the concept behind the intermediate value theorem in calculus. simple connectedness from topology is key to proving there exists a single valued branch of the logarithm function in various regions of the plane. the two dimensional version of connectedness, i.e. winding number, is key to one nice proof of the fundamental theorem of algebra. also producing eigenvalues of a symmetric linear operator T by finding the unit vector x at whose tip the function <x,Tx> is maximized. on a little higher level, the theory of covering spaces from topology gives a nice proof of the embeddability of free groups on many generators inside the free group on two generators. convergence of course is fundamental to many definitions and results in real analysis, hard to pull out a specific nice one just now. maybe using the contraction lemma for complete metric spaces to prove the inverse function theorem. these are really applications to calculus, hard to think of one that impacts real analysis say in the sense of lebesgue integration theory. the stone weierstrass approximation theorem is nice, and the interplay between compactifications of completely regular spaces and the algebras of continuous functions on them. the abstract theory of convergence in hilbert spaces is linked with theory of Fourier series but I am not too savvy about this.

 

[lavinia]

I am not sure what level of Topology you are interested in but here are some classic problems that lend themselves to topological proofs.

- Every continuous function on a closed interval has a maximum and a minimum. Generalize closed interval to compact set.
- Every subgroup of a free group is free.
- The fundamental theorem of algebra (as Mathwonk already said). There are several proofs of this.
- The seven Bridges of Konigsberg - Google it but don't read the answer.
- The Brouwer Fixed Point Theorem - every continuous map of disc into itself has a fixed point. Or if you continuously stir your coffee there will always be a molecule above its original point. A proof using analysis exists but was discovered after the topological proof.
- The Ham Sandwich Theorem - very hard. Google it but don't read the answer.
- Uniform continuity and uniform convergence are extremely important. Examples to work through are space filling curves and the devil's staircase.
- The Law of Biot and Savart has topological properties. If a magnetic field is generated by a current loop then the amount of work that the field does on a second current loop is called the linking number of the loop(up to a constant). What is the difference in work if the two loops are linked or unlinked?
There are also interesting topological properties of magnetic filaments. Find a paper on helicity of magnetic field lines on the Sun.
- Prove that every continuous flow on the sphere must have at least one source or sink. Show that on a torus this is not true.

 

[Avatrin]

I am not sure what level of Topology you are interested in but here are some classic problems that lend themselves to topological proofs.

I am currently trying to learn most of part 1 from Munkres' textbook. The chapters are called:

Set theory and logic
Topological spaces and continuous functions
Connectedness and compactness
Countability and separation axioms
The Tychonoff theorem
Metrization theorems and paracompactness
Complete metric spaces and function spaces

 

[mathwonk]

at least get through connectedness and compactness. those are absolutely fundamental concepts that come up forever. Complete metric spaces are also important. I myself have never used the Tychonoff theorem, and paracompactness is a technical notion that I have only used to verify a Riemannian metric exists or something like that, and it suffices just to know the statements of those things, as well as Tychonoff, in my possibly narrow view. Function spaces are of course important since they are where the answers to problems live.

By the way the absolute most efficient tool for proving things about connectedness is to notice that a space X is connected if and only if every continuous map from X to the discrete set {0,1} is constant. All other properties of connectedness follow easily from this characterization, much more so than from the definition. For instance: if X is connected and f:X-->Y is a continuous surjection, then Y is also connected. proof: If not there would be a continuous surjection Y-->{0,1}, hence also X-->Y-->{0,1} would be a continuous surjection, contradiction. Another one: if X and Y are connected and have a common point p, then XunionY is connected. proof: If not there is a continuous surjection f from XunionY to {0,1}. But f is constant on X and on Y and both constants must equal f(p), so f is constant on the union, contradiction. This generalizes immediately to any union of conected spaces with a common point. Another one: the closure of a connected subspace is connected. you try it.

I don't know about Munkres, which I have never seen, but most books I know stumble around with clumsy tedious proofs using open sets to prove these things.

 

[HallsofIvy]

Topology isn't intended to "solve problems" in the sense in which you mean it. Its purpose is to give a rigorous footing to "continuity" and "limits".

 

[WWGD]

In many areas of Mathematics you may be able to get by with very little Point set topology, which you can just handwave, which nowadays has taken a very distant second place to Algebraic Topology. I think classical areas like analysis make heavier use of it than some of the new, more "Categorical" areas. Still, for an overall training I think it is good to know.

 

[zinq]

In many areas of Mathematics you may be able to get by with very little Point set topology, which you can just handwave, which nowadays has taken a very distant second place to Algebraic Topology. I think classical areas like analysis make heavier use of it than some of the new, more "Categorical" areas. Still, for an overall training I think it is good to know.

I would not generally agree with this assertion. It is crucial to have a very solid grounding in point-set topology (some areas more than others) in order to do topology of manifolds, differential geometry, and analysis. The less important areas might be — usually — non-Hausdorff spaces, infinite products of spaces, and some other little-used areas. As for the overlap between physics and mathematics — currently — maybe WWGD's assertion is true. Yet one should be very comfortable with most areas of point-set toplogy to learn algebraic topology well.

 

[WWGD]

I would not generally agree with this assertion. It is crucial to have a very solid grounding in point-set topology (some areas more than others) in order to do topology of manifolds, differential geometry, and analysis. The less important areas might be — usually — non-Hausdorff spaces, infinite products of spaces, and some other little-used areas. As for the overlap between physics and mathematics — currently — maybe WWGD's assertion is true. Yet one should be very comfortable with most areas of point-set toplogy to learn algebraic topology well.

I actually like pointset. Still, If you look at, e.g., Hatcher's book, you will see that there is barely any use of pointset. Many results, like the continuity of the homotopy maps are brushed of; real pointset issues, like the fact that functions from product spaces may not be continuous even if the restriction maps f(.., x_i,..) are continuous are brushed aside. Similar for Massey, etc. But , I agree if you want to do anything involving analysis (sp. functional analysis) or calculus on manifolds, you do need to know your pointset. A lot of the stuff seems to have reached either a dead-end or just a lack of interest. Barely anyone cares nowadays about Souslin trees, or specialized T_{a/b} space issues, or metacompactness, etc. Even compactification issues are not often mentioned, if not mentioned. I remember reading postings for thesis defenses, not one I remember involved any issues of pointset. The results are used apparently " under the hood". My proxy is that you don't see profs that are dirty with chalk after class -- this happens with the classical subjects where a lot of details/specifics come into play ;) . You see the profs clean, no chalk marks, since they use mostly commutative diagrams and category -theoretic arguments. I personally don't like the modern, over-clean IMHO approach of much of modern Mathematics. The general trend seems to be towards the categorization of Mathematics. Not a judgement, just my take on what I see happenning.

 

[zinq]

I actually like pointset. Still, If you look at, e.g., Hatcher's book, you will see that there is barely any use of pointset. ...

I agree that algebraic topology books rarely mention pointset topology, but this is for the same reason that high school algebra books rarely mention arithmetic: It is assumed the student is familiar with it. Not because it is not used, since it is used all over the place, but tacitly.

By the way, I totally agree, maybe even more enthusiastically, that the "overly clean" approach in mathematics is especially bad when teaching a subject (and when self-styled hotshots write Wikipedia articles). In most math subjects it is absolutely necessary to get one's hands dirty with details and examples (if not chalk) to really understand a subject. That is one reason that Hatcher's algebraic topology book is a thousand times better for teaching the subject than Spanier's.