Recent content by bedi

$\Bbb{R}P^1$ bundle isomorphic to the Mobius bundle I'm trying to construct an explicit isomorphism from ##E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}## to ##T = [0, 1] × R/ ∼## where ##(0, t) ∼ (1, −t)##. I verified that ##\Bbb{R}P^1## is homeomorphic to ##\Bbb{S}^1## which is homeomorphic to...

The best book you can ever find on GR is Gravitation by Wheeler but I am sure you have heard of it (in fact you might have read it).

Yes, this is why I said R x {something} doesn't work. I think I should write these in latex, it may be the case that you think I wrote U as a set but it is the union operation.

(1) A={{x: f(x)=-y^2>-1}x{y'<-1: y' is a real}}U{{x: f(x)=-y^2>-2}x{y'<-2: y' is a real}}U{{x: f(x)=-y^2>-sqrt2}x{y'<-sqrt2: y' is a real}}U... (2) is similar, I just substitute random real values for y.

Actually you can ignore my last reply above. Alright so write {x:f(x)>y} instead. We know this is open because f is continuous. y is defined as an arbitrary real number such that y<sup f so that y<f(x) could make sense. And if we put y'<y we get an open subset of real numbers consisting of y'...

But this doesn't contain those y's. If we add them manually to A will that change something? Edit: If f has a max, say f(D), then add (f^-1(D), y) to A. If not then there exists an f(x') such that f(x')>y so add (f^-1(x'), y) to A. What do you think?

Let A=U_y {{f^-1(x):f(x)>y}x{y':y<maxf(R) is there is such a max and y'<y}}. Does that work?

Let f be a continuous function from R to R and let A be a subset of R^2. Define A={(x,y): y<f(x)}. Can you express A as a cartesian product of two open sets? I tried RxU alpha_x where alpha_x = {y:y<f(x)}. But that didn't work, i need to change something about R.

Of course I did not assume that S is closed. So it is wrong in general that if we subtract (complement) any type of set (closed, open, clopen) from an open set, in this case A, the result is open, yes?

Theorem: Let S be a compact subset of ℝ^n. Then S is closed. Before looking at the book I wanted to come up with my own solution so here is what I've thought so far: Fix a point x in S. Let Un V_n (union of V_n's...) be an open covering of S, where V_n=B(x;n). We know that there is a...

Homework Statement Suppose c_n is the digit in the nth place of the decimal expansion of 2^1/2. Prove that the radius of convergence of \sum{c_n x^n} is equal to 1. Homework Equations The Attempt at a Solution What I want to show is that limsup |c_n|^1/n = 1. Clearly for any...

Alright, that can happen right? Why not?

Thank you, you are so helpful!

On the other hand, mfb asked that "if you want to define an equivalence relation where each class has 9 members, how many equivalence classes do you get?" and since equivalence classes are also a partition on A, we have to have 2^99/9 classes in this case, which is not possible. Correct?

Yes I think this is an equivalence relation on A. And the equivalence classes are: [1]={{1}}, [2]={{2},{3}}, [3]={{3}}. Actually I'm not sure if an equivalence relation on any given set must say something about all the elements of the set. For instance in this eq. relation we do not have [4], is...