I'm going to prove every single theorem in topology!


by mathboy
Tags: prove, single, theorem, topology
mathboy
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#1
Jan25-08, 05:54 PM
P: 182
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?
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wildman
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#2
Jan26-08, 12:24 AM
P: 252
A good place to start is the web page: http://arxiv.org/ They have a section on topology under the math section.
Emmanuel114
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#3
Jan26-08, 01:04 AM
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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
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#4
Jan26-08, 09:46 AM
P: 182

I'm going to prove every single theorem in topology!


Quote Quote by wildman View Post
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
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#5
Jan26-08, 10:12 AM
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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
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#6
Jan26-08, 10:19 AM
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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
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#7
Jan26-08, 10:29 AM
P: 182
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
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#8
Jan26-08, 11:42 AM
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Quote Quote by mathwonk View Post
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
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#9
Jan26-08, 11:44 AM
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Quote Quote by mathboy View Post
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
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#10
Jan26-08, 01:43 PM
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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
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#11
Jan26-08, 02:02 PM
P: 182
This book (Encyclopedia of General Topology)
http://www.amazon.com/exec/obidos/tg...X0DER&v=glance

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
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Jan26-08, 02:07 PM
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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
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#13
Jan26-08, 02:13 PM
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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
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Jan26-08, 02:38 PM
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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
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#15
Jan26-08, 02:43 PM
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Quote Quote by quasar987 View Post
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
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Jan26-08, 07:40 PM
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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), [itex]\beta\mathbb{N}[/itex]. [itex]\mathbb{N}[/itex] is countable, but [itex]\beta\mathbb{N}[/itex] 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 [itex]\beta\mathbb{N}\ \backslash \mathbb{N}[/itex]! 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 [itex]\beta\mathbb{N}[/itex].

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
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#17
Jan26-08, 09:45 PM
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Quote Quote by mathboy View Post
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
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#18
Jan26-08, 09:49 PM
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you guys know this kid is like 15 too


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