I'm going to prove every single theorem in topology

[mathboy]

I've started studying point-set topology a month ago and I'm hooked! I guess one reason is because each question is proof-based, abstract, and non-calculational, which is what I like. I've decided to take on the project of proving every single theorem in topology (that is found in textbooks), starting with the easiest and working my way up.

I own six topology textbooks and will be buying more. Every theorem serves as a topology problem, and I will be trying to prove every single one, peeking at the proof when necessary, and I'm going to write out every proof as well (with no detail left out). The exercises themselves are also theorems which of course I will try to prove.

Now here's my question for you. I will often get stuck trying to prove a theorem, so having the proof in the book will be handy when I have no other choice but to take a peek. For the theorems found in the exercises where there's no proof in any of the other books I have, can I find the proofs somewhere (if I'm stuck trying to do the proof myself). If I can't find the proof in any of my textbooks (because the theorem is very odd-ball), where can I find it? For example, I encountered an exercise that asks me to prove a theorem about orthocompact spaces. I am stuck but can't find that topic in any of my other textbooks. The (original) proof must be published somewhere, so how do I know where I can find it?

 

[wildman]

A good place to start is the web page: http://arxiv.org/ They have a section on topology under the math section.

 

[Emmanuel114]

Dude, you should use your college library if you want to study a subject. Owning 6 textbooks on point-set topology is overkill and bad for your wallet. Diversify your subject matter to other subjects like linear algebra or elementary number theory.

 

[mathboy]

A good place to start is the web page: http://arxiv.org/ They have a section on topology under the math section.

Ah, after reading the proofs in these articles I realize that these odd theorems are proven only once (by the original discoverer), accepted, and probably never repeated again. That's why I can't find the proofs of the odd theorems in my exercises in any of my textbooks. The proofs can only be found in the original paper.

 

[mathwonk]

this is like a man who says he is going to eat all the food in the world, but before starting he wants to collect it all in one pile in font of him. just be content to eat what you have in the frig. you will find plenty proofs you do have access to, long before getting to the problem of theorems whose proofs cannot be found.

and another point is that the non computational aspect of the point set topology proofs you are learning essentially means they are trivial foundational results which follow tautologically from definitions. real math is about computations and concepts combined.

i speak from experience having had a similar delighted reaction to the topology in kelley's book some 40 years ago. later i realized i was very weak, since i had not mastered significant examples and had acquired no mathematical muscle from hard computations.

 

[mathwonk]

just try proving this one result: the unit circle cannot be continuously pulled away from the origin without passing through the origin.

proving the differentiable case of this fact, while teaching calculus of several variables, was my beginning at understanding "algebraic topology".

or, prove that if you connect the two pairs of opposite corners of a square by two continuous arcs, the two arcs must meet. bott gave us this exercise one day in class.

 

[mathboy]

Ok, then I shall prove every theorem and work out every major example and counter-example in topology. Thereby developing both abstract strength and computational strength.

 

[JasonRox]

this is like a man who says he is going to eat all the food in the world, but before starting he wants to collect it all in one pile in font of him. just be content to eat what you have in the frig. you will find plenty proofs you do have access to, long before getting to the problem of theorems whose proofs cannot be found.

and another point is that the non computational aspect of the point set topology proofs you are learning essentially means they are trivial foundational results which follow tautologically from definitions. real math is about computations and concepts combined.

i speak from experience having had a similar delighted reaction to the topology in kelley's book some 40 years ago. later i realized i was very weak, since i had not mastered significant examples and had acquired no mathematical muscle from hard computations.

Good analogy.

Yeah, I think 6 books on Point-Set Topology is overkill.

Also, to add to mathwonk's analogy, when all the food in the world is in front of you, it's hard to tell what's in the middle. You may eat for awhile and see something you can't eat after uncovering it! Make the connection.

 

[JasonRox]

Ok, then I shall prove every theorem and work out every major example and counter-example in topology. Thereby developing both abstract strength and computational strength.

I thought I was setting my standards high when I simply just want to read an Algebraic Topology textbook and understand it well enough to think about the stuff and just enjoy it.

Your standards is just beyond imaginable. Mathematicians collect counter-examples of decades of work and thought and not weeks or years or even in ones own life by himself.

 

[mathboy]

So if a see a rare definition in an exercise and I can't find theory about it in any textbook, how do I find the original paper that introduced the defintion and proved the properties of it? Isn't there some sort of central inventory that I can use to find where the original proof is?

 

[mathboy]

This book (Encyclopedia of General Topology)


has all the definitions and theorems in general topology but gives no proofs. I'm going to take on the task of compiling all the proofs and save it in one big anthology of topology proofs. Of course, I will try to prove the simple theorems myself, but where I'm stuck, I need to know where I can find the original proof of the other theorems that I cannot prove (and can't find in any textbook). So how do I find the proof of a specific theorem in the math journals? Is there some sort a system where I can type keywords and get the proofs?

 

[mathwonk]

when i was a young student i greatly enjoyed the book, by gelbaum and olmstead called counterexamples in analysis. there is also one called counterexamples in topology, by maybe lynn steen. you might enjoy that.

 

[mathboy]

Ok, let me draw upon a specific example so that you understand my project. Currently, I'm compiling all the well-known theorems about T1-spaces (then I'll move on to the next topic). I type out all the major theorems and their proofs (some by me alone, some with help of my textbooks) and save them in one file. A have about 20 theorems completed so far. Currently I'm stuck on this theorem:

A subset Y of a T1-space is countably compact if and only if every countable open cover of Y has a finite subcover.

This appeared as an exercise in one of my topology textbooks. I'm stuck trying to prove it and can't find the proof in any of my other topology textbooks. I don't want to bug you guys for the proof, so if the proof does not exist in textbooks tell me how I can find the proof in the topology journals. Is there a quick way to look it up?

 

[quasar987]

I was under the impression that "A subset Y of a topological space is countably compact if and only if every countable open cover of Y has a finite subcover." is the definition of "countably compact".

What does "countably compact" mean to you?

 

[mathboy]

I was under the impression that "A subset Y of a topological space is countably compact if and only if every countable open cover of Y has a finite subcover." is the definition of "countably compact".

What does "countably compact" mean to you?

Countably compact means that every infinite subset has a limit point. A T1-space is a space where for every two distinct points x,y there is an open set containing x but not the y and vice versa.

 

[morphism]

I also question how good of an idea it is to work through 6 books on point-set topology. It seems to me that everything you need to know can be found by working through Munkres paired off with either Willard or Kelley. Most of the stuff about T_i spaces is mostly dead anyway, research-wise. Maybe your time will be spent better by working through enough preliminary material to get to the main results, like Tychonov, Urysohn, the Stone-Cech compactification, etc. Then, if you have enough classical analysis and measure theory under your belt, you'd be in good shape to tackle functional analysis. There's a lot of topology involved here, but this time it's richer because it's fused with underlying linear algebraic structure.

For example, given a complex normed space X, we know that X is normal so that the space of continuous functions from X to C separates the points of X (by Uryoshn). But now let's take advantage of the linear structure of X - how about we consider the linear continuous functions from X into C - will there be enough of them to separate the points of X? If X is finite dimensional, then this isn't hard to show, but what if X is infinite dimensional? The answer is still yes: this is the Hahn-Banach theorem. Therefore the space X* of continuous complex-valued linear functions on X is large enough, and so we might try to take it apart and see what it looks like. We know that it's naturally a vector space under pointwise operations, but we want to topologize it! There's a very natural way to in fact give it a norm topology through X, but one might notice that X* is a subset of C^X (the set of complex-valued functions on X), and that the latter can be given the product topology (how?). So how about we consider X* as a subspace of the product space C^X? This gives us a nice topology, called the weak* topology (can you guess why the adjective "weak" is used?). Here's where some topological knowledge will come in handy to show how rich the weak* topology is; for instance, we can prove that the closed unit ball of X* (when it's given the natural norm from X) is weak* compact (not surprisingly, Tychonov is the main proponent of the proof). This is really good, because one of the things one loses when they move from finite dimensions to infinite dimensions is the compactness of the closed unit ball.

If hard analysis isn't really what you like, and if you prefer a lot of abstraction, you might consider doing some set theory. Set theoretic topology is really nice, and is currently an active area of research (or so I'm told). Presumably you've seen some examples of giving an ordinal the order topology already. Ordinal spaces give us a wealth of examples of weird topological spaces. Something one can also look at is the Stone-Cech compactification of the natural numbers (given the discrete topology), $\beta\mathbb{N}$. $\mathbb{N}$ is countable, but $\beta\mathbb{N}$ is enormous, having the cardinality of the power set of the power set of the naturals (i.e. 2^(2^aleph0))! That said, we cannot even explicitly exhibit (using ZFC set theory) a point in $\beta\mathbb{N}\ \backslash \mathbb{N}$! As another interesting application, we can get a characterization of X* when X is the space of bounded sequences of complex numbers: X* is isometrically isomorphic to the space of complex-valued functions on $\beta\mathbb{N}$.

And of course, there's always algebraic topology if you know some group theory. Personally I haven't studied enough algebraic topology for the sake of algebraic topology to know anything worth saying about it. But from what I have studied, I can tell that it's beautiful, and has many applications to several fields of math (including analysis!).

What I'm saying is: there's a lot of interesting math out there. Try to sample various things - you don't have to eat it all right now.

 

[leon1127]

Countably compact means that every infinite subset has a limit point. A T1-space is a space where for every two distinct points x,y there is an open set containing x but not the y and vice versa.

I know the definition.

So what does "countably compact" mean? Why is it important to have such definition? In what situation does countably compact actually come up? If you can't answer, that means you can do proof, but you don't understand topology.

 

[ice109]

you guys know this kid is like 15 too

 

[morphism]

Is he?

 

[leon1127]

you guys know this kid is like 15 too

In other post, he/she mentioned that he is college freshman. https://www.physicsforums.com/showthread.php?t=207213"

Mathboy, I am not trying to discourage you from doing problems. However, it is more important to know what you can do with all the mathematics you learn instead of doing problems pointlessly.

 

[mathboy]

My goal right now is to improve my problem solving ability, and proving topology theorems would help me in that regard. I was the person who nominated Morphism in the lounge to get the Homework Helper recognition because I was so impressed by his problem solving ability and his ability to solve problems correctly in every subject. Right now I am still low in the knowledge category, but I figure that if I prove my problem solving skill first, then I can develop knowledge quickly.

Morphism, I just want to ask you how you developed the ability and how you retain your knowledge of every theorem and problem solving ability in every subject? I read one of your threads where you panicked over a final exam so I gathered that you are still a student (perhaps a graduate student now). So far, I see that you are very versed in calculus, linear algebra, real analysis, functional analysis, set theory, point-set topology, group theory, ring theory. What else is there? And how do you retain all that?

 

[mathboy]

I just finished proving 27 theorems on T1-spaces, and will now move on to T2-spaces now. Only one theorem I could not prove:

A subset of a T1-space is countably compact if and only if every countable open cover of the subset has a finite subcover.

I will gladly share my work with you guys.

 

[morphism]

I am still a student - an undergraduate too (although I'm almost done). I've found that the best way to retain information is to use it constantly. What better way to do this than to participate in a forum where the material is always discussed, like this one? (Thanks for your compliments and nomination, by the way. )

If you're goal is to hone your problem solving skills, then what you're doing will probably do that and much more. It's very impressive that you're doing topology as a freshman, but I second leon1127's concern: try not to lose sight of the big picture. One time I was discussing something with a very bright friend of mine - it was related to weak and strong convergence in Banach spaces - and my friend knew the definitions, and could prove the main theorems and solve the difficult problems. But when I asked him to give me an example of a weakly convergent sequence that wasn't strongly convergent, he couldn't! So to me it seemed like he was missing the point, and not really thinking about the material he's learning, even though it seemed like he knew it very well.

For your problem, let's try the following. Call the space X, and let A be a subset of X. If A is finite, there's nothing to prove, so let's assume that A is infinite. Suppose A is countably compact. Let U={U_n} be a countable open cover of A. Let S consists of all the points in A that are missed by some finite subcover of U. If S is finite or empty, then U admits a subcover that covers A (why?). On the other hand if S is infinite, then it has a limit point a_1 in A - say a_1 sits in U_1. Now, being an open nbhd of a, U_1 intersects S. If S\U_1 is infinite, then it has a limit point a_2 that sits, renumbering the cover if necessary, in U_2\U_1 (why can't a_2 be in U_1?). If S\(U_1 union U_2) is still infinite, then it has a limit point a_3 that sits in U_3\(U_1 union U_2). Keep doing this to get a_4, a_5, ... If the set {a_n} is infinite, then it has a limit point a_$\infty$. Now a_$\infty$ sits in some U_k, but U_k doesn't contain a_(k+1), a_(k+2), ..., so that U_k intersects the set {a_n} at only finitely many points. But this is impossible, because we're in a T_1 space! (Prove: In a T_1 space, any nbhd of a limit point of a set S contains infinitely many points of S.) So there must exist an N for which S\(U_1 union U_2 union ... union U_N) is finite, and from this we can again get a finite subcover for A.

Conversely, suppose every countable open cover of A admits a finite subcover. Suppose B is an infinite subset of A that has no limit points. Thus, for each b in B we can find an open set U_b that intersects B exactly in {b}. In particular, we can do this for any countable subset {b_1, b_2, ...} of B. But then if we let U_n=U_(b_n) for each n>=1 and U_0=A\{b_1, b_2, ...}, the open cover {U_n : n>=0} of A will admit no finite subcover (why? and why is it an open cover?).

How does that sound?

 

[mathboy]

Wow! I'll try to fill in the gaps that you've intentionally placed.

If S is finite or empty, then U admits a subcover that covers A (why?)

because S can be covered by a finite subcover

why can't a_2 be in U_1?

because a_2 is a limit point of S\U_1 , so if a_2 is in U_1 then U_1 is a neighbourhood of a_2 not intersecting S\U_1, a contradiction

Prove: In a T_1 space, any nbhd of a limit point of a set S contains infinitely many points of S.

This is one of the 27 theorems. Already done.

Let S consists of all the points in A that are missed by some finite subcover of U.

You mean "Let S consists of all the points in A that are missed by some finite subcollection of U, right?"

 

[morphism]

Yup, that's what I meant. Does everything else look good? I posted my attempt for the other direction. Initially I was running into trouble because I kept getting to a point where I thought I needed a metric, and that's because I wasn't thinking about "limit points" correctly!

 

[mathboy]

Now a_$\infty$ sits in some U_k

Not every member of the cover U needs to be occupied by an a_i term. A countable set can map non-surjectively to the positive integers.

and U_0=A\{b_1, b_2, ...}, the open cover {U_n : n>=0} of A

{b_1, b_2, ...} is not a closed set (yet). In a T1-space, finite sets are closed, but {b_1, b_2, ...} is not finite. However, B itself is closed because B contains all of its limit points (since it has none). Then {b_1, b_2, ...} = B n {b_1} n {b_2} n ... is now closed.

 

[morphism]

a_infty sits in A, which is covered by U. So it must sit in one of the elements of U.

 

[mathboy]

I know that, but by definition U_k is the member of U that contains a_k. What I'm saying is that a_infty does not have to be in a member of U that is occupied by an a_n term, right? In that case saying that "U_k doesn't contain a_(k+1), a_(k+2), ..., " doesn't apply.

 

[morphism]

But by construction, if {a_n : n>=1} is an infinite set, then each U_k will contain an a_k. This is because at the kth point of the construction, we're dropping the first k-1 elements of the cover, namely U_1, ..., U_(k-1), and choosing an element a_k that doesn't lie in any of them (since taking the union of open sets will give us an open set). This a_k will then sit in one of U_k, U_(k+1), ..., and we can assume WLOG that it's in U_k.

Edit:

{b_1, b_2, ...} is not a closed set (yet). In a T1-space, finite sets are closed, but {b_1, b_2, ...} is not finite. However, B itself is closed because B contains all of its limit points (since it has none). Then {b_1, b_2, ...} = B n {b_1} n {b_2} n ... is now closed.

Spot on!

 

[mathboy]

each U_k will contain an a_k.

This is the only point I question. (everything else is perfect, including the converse)

Picture the open cover U as an infinite checkerboard with red and black squares. Why can't all the a_n be in the black squares, and a_infinity be in a red square?

 

[morphism]

By construction, a_n will fill up the entire checkerboard.

If you don't really like the juggling of indices, I suppose we can go in the 'reverse' direction. As before, let U={U_n} be a countable open cover for A. This time suppose no finite subcollection of U covers A. Then there is an a_1 in A not covered by {U_1}, and an a_2 (that is different from a_1) in A not covered by {U_1, U_2}, ..., and an a_n (that is different from a_1, ..., a_(n-1)) in A not covered by {U_1, ..., U_n}, and so on. So we have this infinite subset A'={a_1, a_2, ...} of A that cannot possibly have a limit point; for if a_infty is a limit point of A', and a_infty sits in A_k, then A_k is an open nbhd of a_infty that can only intersect A' in {a_1, ..., a_(k-1)}.

 

[mathboy]

Ok, now the proof leaves no doubt! Excellent, I've uploaded all the my worked out proofs regarding T1-spaces in post #22, and I accredited the proof to this theorem to Morphism.

 

[rudinreader]

I should have read this thread before starting the thread https://www.physicsforums.com/showthread.php?t=211744.

wonk brings up an extremely important point:

and another point is that the non computational aspect of the point set topology proofs you are learning essentially means they are trivial foundational results which follow tautologically from definitions. real math is about computations and concepts combined.

i speak from experience having had a similar delighted reaction to the topology in kelley's book some 40 years ago. later i realized i was very weak, since i had not mastered significant examples and had acquired no mathematical muscle from hard computations.

In other words - if you are a student that spends 6+ months on a general topology or even worse a "set theory" book without even having nailed down basic analysis, then you are likely not going to gain that much. On the other hand, if you spend 6 months SUCCESSFULLY solving problems in a book like Rudin's Principles of Analysis, you are likely to gain a LOT.

Another person mentioned some pertinent comments on the website http://math.stanford.edu/~vakil/potentialstudents.html on the thread https://www.physicsforums.com/showthread.php?t=209716


So given that, how can I justify doing topology before "rigorous Calculus"? The answer is: don't spend that much time on it. On the other hand, the learning curve to the "epsilon-delta" is steep. The "completeness axiom" of R is ultimately required, and by describing mathematical "connectedness" and pointing out that the ordered field axioms don't imply connectedness, we can give a concrete motivation for the completeness, and perhaps that could motivate the epsilon-delta which is really most important..

Anyways, just an idea. Every once in a while I get side tracked on little things like this, but I would prefer to finish up today and get it overwith and get back to selfishly doing math for myself.

 

[mathwonk]

again, it is much less useful to know what a T2 space is than to know about lens spaces, projective spaces, grassman spaces, vector bundles, manifolds, covering maps.

 

[rudinreader]

So given that, how can I justify doing topology before "rigorous Calculus"? ...

Actually I would like to add, that I'm not actually trying to justify anything. This is more of a thought exercise, "if I were to present general topology before Calculus, how would I write it up?" And ultimately, it was probably a dumb thought exercise to begin with..

But I'm sure something can be learned from me posting - if nothing else a good example of a bad choice of a problem to pursue,..