I'm going to prove every single theorem in topology

[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,..

 

[masnevets]

To be honest, I haven't spent much time learning point-set topology. That seemed largely irrelevant when I took a course in (graduate) algebraic topology. I just learned what I needed to learn as I went along.

At some point after I decided I would go through Munkres' _Topology_ and do every single exercise to fill in the gaps. But I got bored with it very quickly after it seemed that most of the point-set topology that I didn't already know would probably never be useful to me.

From my studies, I've really only needed a few things about point-set topology like "X is Hausdorff iff its image in the diagonal is closed" and what does connected and compact mean. I agree with mathwonk that it's largely unimportant if you know what properties T3 and T4 spaces have (I can't even define them).

If you want to build problem solving skills, read different types of math books. Your mind isn't going to be challenged otherwise.

 

[andytoh]

To be honest, I haven't spent much time learning point-set topology. That seemed largely irrelevant when I took a course in (graduate) algebraic topology. I just learned what I needed to learn as I went along.

At some point after I decided I would go through Munkres' _Topology_ and do every single exercise to fill in the gaps. But I got bored with it very quickly after it seemed that most of the point-set topology that I didn't already know would probably never be useful to me.

From my studies, I've really only needed a few things about point-set topology like "X is Hausdorff iff its image in the diagonal is closed" and what does connected and compact mean. I agree with mathwonk that it's largely unimportant if you know what properties T3 and T4 spaces have (I can't even define them).

If you want to build problem solving skills, read different types of math books. Your mind isn't going to be challenged otherwise.

In my opinion, one should become versed in set-theoretic topology first before specializing or even taking algebraic topology or differential topology. You could still manage, but I think you would learn much more easily and do better if you mastered your reading and proving skills in set-theoretic topology first. When I first studied differential topology, I had a hard time proving that the projective space and Grassman space is a manifold because I was rusty with quotient spaces. When I later studied quotient space and did problems in it, I realized I would have been able to understand those manifolds instantly had I studied quotient topologies beforehand. Same goes with understanding that every atlas was contained in a maximal atlas, because I was not versed enough in Zorn's lemma at the time. I later realized I should have studied set-theoretic topology first before studying differential topology.

 

[rudinreader]

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.

OK Mathboy I read your comments and I guess you (and perhaps other young freshman/high school students) want to get a head start on sophisticated math..

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.

Actually, if you really want to collect topology in a pile, you can do it, but you should still focus on one book. Probably G. Bredon Topology and Geometry. It's a big book for well prepared first year graduate students, supposedly 2 full years worth of topics to feast over.

But as a freshman, you will probably find that the style of writing is hard to follow because of the prerequisites assumed. But from what I read of the book, it seems the only main prerequisite is a solid grasp of undergraduate analysis at the 4000 level - such as Rudin Principles of Analysis.

In summary, having the ability to work through the abstract ideas as in this thread should be most helpful for you if you are going to study 4000 level analysis. It is more likely then that the early problems (in the problems to chapter 2) in Rudin concerning showing that compactness is equivalent to every infinite subset having a limit point, will be more likely to stick whereas a lot of people who read the book don't catch the point.

If you really grasp what is inside the (any good) 4000 analysis book (which if you can forgive personally that I did a bad job reading L'Hospital), then you can go ahead with Bredon's. Plus - it helps to have at least heard of a lot of other basic things, like knowing every linear transformation is a matrix in the finite vector space, etc.

Then if for some really strange reason (that I wouldn't personally understand) you find that Bredon is more interesting than green Rudin, then you could have a very fruitful (long term) study of a large subject..

But I have found from experience if you want to pursue this kind of project it is best to focus on one book, unless you are just (weakly) surveying some books for theorem statements in a prelim exam.

And don't mistreat 4000 level analysis - it's not really the theorems per se (which often amount to the theorems of Calculus), but the mathematical maturity to be developed... As I always say, if you are already good at it, you could easily get through the book more quickly.. If you are not good at it, then you won't get much out of topology or analysis (but perhaps you can get by studying a book like Knuth's "Art of Computer Programming", and that's a big "perhaps"!).

I would also mention that people are always commenting on how important linear algebra is, and it is, but ultimately I have found that reading and solving the problems out of a book like Hoffman/Kunzes Linear Algebra is no where near as strong of an exercise as Rudin because there's no epsilon-delta involved. So if you had to choose, you don't even need linear algebra at all as a prerequisite, except ultimately those theorems are used everywhere (it's especially assumed you can make a linear algebra argument in Bredon's Topology) as well so you'll eventually become familiar with it.

ANyways, that's my thoughts (probably just B>S>)

 

[biouser]

I will be trying to prove every single one... I'm going to write out every proof as well (with no detail left out).

I like your pluck. I am writing a web application for math writing and math writing collaboration (trying to be like a math-only, more vetted wikipedia-ish beast). The code is open source (not posted yet though) and it will all be free and so forth. A full distribution of LaTeX will drive the rendering (with normal LaTeX syntax). The operating name right now is:

PyWebMath

Once I have it up and the database schemas are set in stone it would be great if you could try to concurrently accomplish your goal with the site. If you are writing in LaTeX you could just paste into the field. You could be the only one privileged to edit your posts or you could add people to your group.

Happy Proving!

 

[maze]

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.

Can you please elaborate? My reading of this seems like its not true.
http://img201.imageshack.us/img201/2289/squarearcskz7.png

 

[Pere Callahan]

My reading is, the arcs must stay inside the square.

 

[maze]

My reading is, the arcs must stay inside the square.

Then it would be a straightforward consequence of the jordan curve theorem, right? (a non-intersecting loop divides the plane into 2 disjoint parts). Assume the the curves have no intersection, and then construct 3 different nontrivial loops that split the box into distinct pieces. In the set inclusions/exclusions there will be a contradiction (probably).

 

[Werg22]

I think he meant circle arcs.

 

[maze]

I think he meant circle arcs.

You mean, like an entire circle?

 

[Werg22]

Simply two circles arcs, each connecting one pair of diagonally opposite points, i.e., by construction, take a compass and find a point equidistant from a pair of diagonally opposite points, place the pivot end of the compass on that point and trace the arc joining the diagonally opposite points (you don't have to draw the full circle), repeat the process with the other pair. Regardless of the centers you choose for the arcs, they should meet.

 

[mathwonk]

a continuous arc in the square connecting two opposite corners, is a continuous map f from a closed bounded interval [a,b] into the product [0,1] x [0,1] such that f(a) is one corner, say (0,0), and f(b) is an opposite corner, in this case (1,1).

the jordan curve theorem does not seem to apply at all since no injectivity is assumed for f.

and by the way euclid omitted any axiom sufficient to prove that two circles meet even if one circle has center outside the other circle, and also contains a point inside the other.

 

[maze]

the jordan curve theorem does not seem to apply at all since no injectivity is assumed for f.

The curve would be path-connected though, so all we have to do is show that you can construct a non-self-intersecting path from point to point in a path connected space. I'm not exactly sure how to do this, perhaps it is just another way of stating the original problem.

 

[maze]

Ahh yes, here we go: the image of f is path connected, so it is connected. Since this image is also compact, it is equivalent to some n-torus (R2 "doughnut" with n holes in it/connected sum of n loops). Thus it partitions the plane into at least 2 components (in fact, n+1 components...) as required.

Then apply the rest of the argument.