@TedShifrin Sanity-check : It is true that if X is a CW complex, with homotopy equivalent subcomplexes A, B, then X/A and X/B are also homotopy equivalent, right?
I worked on it on and off for a while, probably about a full year in total. I think with concentrated study it shouldn't take more than 3 months. I did most of the exercises. Part of the reason the book is so good is the exercises; I wouldn't suggest bothering with it unless you do them,.
Yeah, I gave myself 6 weeks to do it. I'm halfway through but have been slacking on the exercises :/ some take longer than expected and I end up having to develop lots of lemmas for them.
There is no continuous function that does what you want, @Hippa. There are lots if you're just demanding a function. e.g., pick your favorite bijection $[0,1/2) \to [0,1]$; and define your map to be that for $x < 1/2$, and $2x-1$ for $x \geq 1/2$.
@TedShifrin You wouldn't have (still the same question) found a function f:[0,1]→[0,1] so that ∀0≤x≤1,f−1(x) is a tuple (2 elements) by any chance ?
Could one give me a bijection between $[0,1]$ and $[0,1)$ ?
I am trying to find a function $f:[0,2]\to[0,2]$ such that $\forall0\le x\le2,f^{-1}(x)$ is a tuple (2 elements), and such a bijection would give me an answer easily.
@JasperLoy You think that problem of yours blocks you from making progress in all the areas of your life? For instance, does it prevent you from going jogging? Maybe not. I might try to improve in those areas where I'm able to improve with the resources I have.
@Chris'ssis where will you publish your book? (in which edition house, e.g. springer) or will it just be createspace.com like the book elementary applied topology, see math.upenn.edu/~ghrist/notes.html , and Sir Ghrist gives details to how much you can win this way
@TedShifrin ah, i see now. i actually had that answer before, but i miscalculated and thought it was outside of the bounds. thank you for the assistance. :)
Oh and, the question before (same exercise) asks to show that $\{k/(2^n)|n\in\mathbb{N},k\in\{0,\dots,2^n\}\}$ is dense in $[0,1]$ but I don't see how to use that
@Chris'ssis but you should most probably find a good graphic design professional who could make for you an excellent cover as well as an excellent overrall formatting (with Tex using tikz-pgf, asymptote, etc...)
@TedShifrin Not in maths, at least (but remember, I'm in one of the 3 best prep schools, not all are like that. Besides, we don't have to do all the 90 exercises, our teacher just gives us some sheets of exercises per chapter, and we have 4h/week where we correct in them in class (students go to the board) with usually half the people in the class. What is more, I'm by far more interested in maths than the average student in my class (I'm in PC Physics-Chemistry, not MP Maths-Physics))
I tell undergrads not to go to grad school in math unless they feel like they cannot live without it, @JohnDoe ... It's just not a good thing to do because you can't think of anything else to do :P
@MathyPerson: I certainly don't like talking about $\tan X$, as $X$ is a point on a ray. You should have angles $\alpha$, $\beta$ from the $x$-axis to points $X$, $Y$ respectively
Has anyone encountered the convention in any book or notes of function spaces $C^{0}(\bar{\Omega})$ and $C(\bar{\Omega})$ being defined as different spaces?
@TedShifrin Oh, that was a typo on my part. It should have said "Angle 1" instead of X and "Angle 2" instead of Y, but alpha and beta can be used instead
@TedShifrin I still don't get what you hint me at. Let $0\le z\le1,z=\lim k_n/2^{n+1}$. The only thing I can think of to use the property is $f(k_n/2^n+k_n/2^n)\le\dots$, but I can't really use the RHS (it involves $f(k_n/2^n)$)
@user153330 You don't know me, I don't know you, so, I wanna tell you a thing: I don't give a f**k on any cent I can get from those books, I mean my only aim is to publish a very good book, at least, (and, yes, I need money, but I simply cannot think of making money by publishing math books). I mean my aims are far more noble, it's something related to my past, it's about promises, about things I cannot share entirely here but they are very important to me.
@Hippalectryon You'll have a book from me for sure if everything goes well (I'm afraid the book won't be at your level that is much higher than mine I suppose ...) :-)
@TedShifrin Well, word for word, it is: Point X is the terminal point of $\alpha$ and point B is the terminal point of $\theta$. Point X is in the first quadrant and point Y is in the second quadrant, while $\tan \alpha = 1$ and $\tan \beta = -7$. Find the slope of $\overline{XY}$.
So, knowing the tangent values, you can certainly find (in terms of square roots, if necessary) the coordinates of both $X$ and $Y$. What's wrong with that?
Alternatively, you know the base angles of $\triangle OXY$, so you can find the angle that $\overleftrightarrow{XY}$ makes with the $x$-axis.
@Chris'ssis so publish it in springer then, you'll have a better marketing overall and your audience will be able to localize your book better than if you self-published it
@user153330 To get a best seller one probably needs to use the language of Paul J. Nahin in his book! I mean the cold mathematical language will never beat his language, that is clear, at least for the large audience.
OK. So when you have both points $X$ and $Y$ on the unit circle, tell me. I actually like other ways of doing the problem better, but this is probably best for you.
@MaryStar: You need to start working with your professor on your homework. This material is highly dependent on exactly how it is presented in the book or in class, and we cannot know that.
@mikeonly there is newtonian classical mechanics, analytic mechanics (lagrangians, hamiltonians, paths..)
@Chris'ssis forgot to mention it, if you want to publish for springer then you may want to send them (the editors of a series) 2 chapters and see their feedback
@user153330 The content of 2 chapters put in the format of a book or the only thing that matters is the content (no matter the format - of course, using latex)?
It's the MIT text for the serious course that uses multivariable calculus, @mikeonly. I took it from the authors as it was being written. Fabulous course.
@user153330 It would be nice if I'd prepare the content in latex, and then they'll create the book. I'm not good at creating books in latex, no experience here!
@Chris'ssis: It took me three books to get really good at it, but my first book was produced (indeed, I had it printed) from my LaTeX. I don't like it anymore :P
I understand, @mikeonly. Here we say, "a mile wide and an inch deep." :P
Could one give me a bijection between $[0,1]$ and $[0,1)$ ?
I am trying to find a function $f:[0,2]\to[0,2]$ such that $\forall0\le x\le2,f^{-1}(x)$ is a tuple (2 elements), and such a bijection would give me an answer easily.
