Good grief. I would never teach Dedekind cuts in a first analysis course. It's absurd. And the students don't appreciate the subtleties. Just use the damn LUB axiom and do some actual interesting stuff.
@shintuku It depends on how you want to approach things. The reals are the unique ordered field (up to isomorphism). If you build the reals using Dedekind cuts, you get the least upper bound property for free. If you build it from Cauchy sequences, you get the Archimedean property for free (more or less). Or something like that---I haven't gone through the details in a long time.
right, but say, i couldn't understand why $sup(A) + sup(B) = sup(A+B)$ needed to involve inequalities. i would think: well this point here on the real line is $sup(A)$, and this point there is $sup(B)$, so clearly we have the equality, why are inequalities involved?
right, but the insight was that a real number is not a point on a line. for any attempt to give a real number, there is a better attempt: that was the real insight for me
@shintuku real number is not a point of line. But we visualize it as a point. ( it's a one dimensional vector space over R. A one dimensional vector space can be thought as line and members as a point on that line) may be a justification but won't make any justice with your initial thought.
I took analysis at MIT out of baby Rudin (early edition, which still did Dedekind cuts). The professor did all that. Never drew a picture the entire course. The worst course of my undergraduate career. Many of the students in the course were taking analysis for the third time. Many took it a fourth time.
Just use either the LUB axiom or the monotone sequence lemma as an axiom and actually do some mathematics. Being pedantic to make yourself feel important is terrible, terrible teaching.
if that information is provided, it also arrives at a time where it's almost misleading to say that the real numbers "are" some construction. students are going to think that the model matters in ways that it doesn't.
Oh, I just looked at the preface to Tao's book. This was for HONORS UCLA students. That's not the general audience for the course. Similarly, Pugh's book was for HONORS Berkeley students. Abbott is much more aimed at the generic student.
This is why the general math major in the US no longer even takes a real analysis course. Pitiful, but true. There are so many different amalgamated/applied majors now.
Well, if you're in the honors course, you should be figuring out a lot more stuff for yourself and not asking us why the proof of a converse isn't adequate to prove a theorem.
The second part is to prove that: assuming that every map $S^1\to X$ can be extended to $D^2\to X$, we must have $\pi_1(X,x_0)=0$ for every $x_0\in X$.
dedekind cuts are great for some purposes, but the definition of multiplication really sucks. even rudin can't deal with it and knows it sucks. you can sense him thinking "oh, forget this, i'm rudin, i can just say verify the cases" while giving a casewise definition of multiplication that looks an awful lot like it's assuming various field axioms (which it isn't, but haha wow it sucks)
koro you can use the D map to write down specific homotopies between the loop based at a point and the constant loop at that point. draw a few pictures.
Interestingly, I had to do this exercise in an undergraduate differential topology course long before I took algebraic topology. So then there's the need to use bump functions to smooth things out at the origin.
So here is how I would go about it step by step: Take $[f]\in \pi_1(X,x_0)$. f: I-->X. $I/\sim \cong S^1$ with the quotient map $q$ so there exists a unique $\tilde f: S^1\to X$ such that $\tilde f\circ q=f$. We extend $\tilde f$ to $D$ (possible by hypothesis) and still use the same notation.
No, you don't need to think about deformation retraction, although I suppose you can. You want to define a map on the disk, so you want to define it circle by circle.
For that I define $g_t: D^2\to X$ as follows: $g_t(x,y)= \tilde f(tx, ty)$ (so I am just taking every point on D^2 to centre (0,0) and then composing it with $\tilde f$.
note that the conclusion is something that needs to be shown for all maps S^1 into X, not just ones that include the basepoint x_0 in their range. i'm not sure what magic is expected to be performed by q.
WinCo is employee owned, unionized, and (usually) rather cheap. Generally very good on bulk food. Produce can be somewhat hit or miss---on a good day, it's excellent, but on a bad day, it can be very unfortunate. The only downside is that they don't take credit cards (cash, checks, and debit cards only).
1.1.6: We can regard $π_1(X,x_0)$ as the set of basepoint-preserving homotopy classes of
maps $(S_1, s_0)→(X,x_0)$. Let $[S_1,X]$ be the set of homotopy classes of maps $S_1→X$,
with no conditions on basepoints. Thus there is a natural map $\phi :π_1(X,x_0)→[S_1,X]$
obtained by ignoring basepoint...
koro, in your MSE question, why would you expect g or g circ q or whatever to be based at x_0? you can certainly consider the homotopy class of any map, but why is it in pi_1(X, x_0) so that you can apply phi to it?
School shootings (and other mass shootings) are a hugely exaggerated problem. According to Pew, there were 45,000 firearm related deaths in the US in 2020. Even using the more liberal estimates, only around 500 of these were in mass shootings. The majority of these deaths were suicides.
Mass shootings are dramatic, but they are not the real problem in the US. :/
@robjohn Indeed. They do, because they are unusual and dramatic. But your chance of being involved in such an incident---let alone being injured or killed---are close to zero.
@robjohn Indeed. I am am genuinely not trying to diminish the shock of it---a colleague of mine was shot and killed by a 12 year old in 2013 (en.wikipedia.org/wiki/2013_Sparks_Middle_School_shooting). After finishing my secondary ed credential, I worked for a bit as a substitute---Michael was one of the people whose classes I taught.
@Koro Because that would require a constitutional amendment, and there is a very powerful, vocal contingent in the US which is opposed.
maybe at some point they'll decide that every one of us needs to carry a gun at all times. or that the constitution requires the government to issue guns to everybody at no charge. i don't want to give them ideas.
Hi! Is there a clever way to count this? 6 red, 5 blue and 7 yellow balls should be placed in six distinguishable boxes, such that every box contains three balls. Balls with same color are indistinguishable.
koro: how does the circle 'of a mobius strip' differ from a circle? (a mobius strip regarded as a surface also retracts onto a circle, so has the same fundamental group as a circle)
however, the inclusion of the boundary circle of the Möbius strip into the Möbius strip is not a homotopy equivalence even though both spaces are homotopy equivalent to $S^1$
i guess how you "see" that Z * Z isn't abelian depends on how you construct or define it, but it should fall out pretty quickly. the vibe is, there's no reason for the generators of the factors to commute. there are homomorphisms of that thing into nonabelian groups that map generators of each factor to whatever elements you like
why would you use VK, it's homotopy equivalent to a circle
which is actually notoriously not something you can compute easily using VK unless you prove the more general version of VK involving groupoids/multiple basepoints (Ronnie Brown loves this example)
the square deformation retracts onto the horizontal line in the middle and this induces on the quotient a deformation retraction of the Möbius strip onto its center circle
you contract all the "vertical" portions and are left with a circle
Ah, but then fundamental group of a torus will also be Z.
Z*Z is not isom. to Z.
so this is wrong.
@leslietownes He first did contraction in square. Then how did identifications happen? contracting a square to a line makes left and right side of the square identification in opposite order meaningless. doesn't it?
I mean the order is lost in this process.
It won't make sense to talk about joining two ends of a thread in opposite order.
to be clear, i was talking about thorgott's deformation of the moebius strip onto the 'horizontal line in the middle' i.e. if the square is IxI with left and right edges identified and the circle is the line segment y = 1/2 in this square. not the torus or klein stuff
Anyone here feel confident in a recommendation of a Measure Theory book? My friend has been nagging me that something therein will help me with a few problems I've been puzzling over.
i don't know of one. measure theory tends to be treated in textbooks as an adjunct to other things because if you just study it by itself it's boring. and the other books it appears in, are not so much good books about measure theory, but good for their other topics.
but, bartle had a book called something like 'integration,' short (maybe 100 pp?), that i liked, for a lot of the basics. from the 1960s. he has some more recent book with a similar title that has more stuff, which might be an expanded version or something different.
i would dis-recommend the rushed treatment in rudin's 'principles of mathematical analysis.' it might even be worse than PMA's multivariable calc section.
yeah, what are you interested in? if you're interested in stochastic processes, that might be a different world from, say, someone who only wants to work on R^n or topological groups.
@XanderHenderson I haven't the foggiest. I'm handling a problem involving expected values and probabilities and I've been tackling them using converging infinite sums, he's a former mentor and professor of mine so he's fond of throwing me at things that primarily expose me to the field and secondarily, meaning by happenstance, may help me out as an alternative method.
For more context, I'll give the simplest example problem I've done so far: Suppose you roll 1d6. If you roll a 6, you may roll again. Determine the expected value of the number of rolls for this situation. You'll find the solution isn't egregious, but then the problem grows in exciting ways when you try for 2d6 or 3d6, where multiple 6's each grant additional rerolls of all the d6s.
He says that there are principles within the field that can lead to faster computations of what I'm tackling. I'm well aware this is a discrete problem, which is also why I was curious behind his suggestion.
@TedShifrin Been there, done that, got the t-shirt and teach the class on occasion. I'm just curious. I've just been abusing general sequence and series techniques to get what I wanted, and I'm sure to higher minds that just seems barbaric.
When he suggested it and another professor chimed in that it was "the logical next step" for what I was doing I wondered if they were just trying to egg me into another textbook.
it's often helpful to approximate complicated or limiting cases of discrete stuff by continuous stuff, but even that tends not to need too much measure theory for things like die rolls. just normal integration on subsets of R^n would be enough.
@TedShifrin He did mention something about an indicator function, and scrawled $\intf(x)dP(x)$ or something of the sort. Admittedly on that part I was juggling students for the evening.
whittle's "probability via expectation" is a pretty in-depth book that never does measure theory as such. some of our favorite functional analysts do similar things to indirectly construct measure spaces
@leslietownes Not likely, he often just points me in a direction or suggests a book I already have access to. This is the first time he actually didn't know of a book to recommend.
i mix caffeinated stuff (usually soda but sometimes tea) with decaffeinated hydration (water or herbal 'tea') all day. i figure as long as the ratio of decaf to caf is high enough, i'm not frying my nerves too much.
Indicator functions tend to just get tossed in at the beginning of an undergrad probabillity course when talking about moving from bernoulli to a binomial count...... Here's this nice $I(x)$ that does this with value $0$ and that with value $1$........
Are these suppose to be the limits of integration to the intersecting planes we were discussing yesterday? $\int_0^1 \int_{1-z}^1 \int_{0}^{2-y-z}dxdydz$......I realized that the lengths of all the sides of the cube were supposed to be $1$ which means that the plane looking down on the cube is actually not at $1$ @TedShifrin
