Homeomorphism Definition and 76 Threads

Homework Statement Let (X,T) be a topological space and let U denote the product topology on X x X. Let delta = {(x,y) in X x X : x = y} and let U_delta be the subspace topology on delta determined by U. Prove that (X,T) is homeomorphic to (delta,U_delta) The Attempt at a Solution...

Homework Statement I have some problems with understanding these two things. Homeomoprhism is a function f f: M\rightarrow N is a homeomorphism if if is bijective and invertible and if both f, f^{-1} are continuous. Here comes an example, let's take function f(x) = x^{3} it is...

Determine the homeomorphism classes of compact 3-manifolds obtained from D^3 by identifying finitely many pairs of disjoint disks in the boundary? I just started reading some low dimensional topology on my own and I came across this question. I have realized that based on how the...

I realize this is a classic problem, but I'm not sure exactly how to start on it: Show that the closed unit square region is homeomorphic to the closed unit disc.

Homework Statement Here's another problem from Munkres. Let (a1, a2, ...) and (b1, b2, ...) be sequences of real numbers, with ai > 0, for every i. Define h : Rω --> Rω with h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...). Show that if Rω is given the product topology, h is a...

Homework Statement If H:I\rightarrowI is a monotone and continuous function, prove that H is a homeomorphism if either a) H(0) = 0 and H(1) = 1 or b) H(0) = 1 and H(1) = 0. Homework Equations The Attempt at a Solution So if I can prove H is a homeomorphism for a), b)...

The definition of a homeomorphism between topological spaces X, Y, is that there exists a function Y=f(X) that is continuous and whose inverse X=f-1(Y) is also continuous. Can I assume that the function f is a bijection, since inverses only exist for bijections? Also, I thought that if a...

Hi Guy's I need to show that two spaces are Homeomorphic for a given function between them. Is there an online example of a proof. A lot of text on the web tells you what it needs to be a homeomorphism but I not an example of a proof. I just want an good example I can you to help me...

hello, im trying to get a homeomorphism between a n-dim vector space and R^n that is independent of the basis. (im actually defining a C \infinity structure on V) since i want a homeomorphism, i know i should define a topology on my vector space, which is the norm, since that would be...

im trying to get a homeomorphism between a 1-dim vector space and R, but independent of the basis. Any ideas?

when I first learned about homeomorphic sets, I was given the example of a doughnut and a coffee cup as being homeomorphic since they could be continuously deformed into each other. fair enough. Recently I heard another such example being given about diffeomorphisms: "Take a rubber cube...

Define the mapping torus of a homeomorphism \phi:X \rightarrow X to be the identification space T(\phi)= X \times I / \{ (x,0) \sim (\phi(x),1) | x \in X \} I have to identify T(\phi) with a standard space and prove that it is homotopy equivalent to S^1 by constructing explicit maps f:S^1...

If D^n is the unit n ball in Euclidean n-space. i.e. D^n = \{ x \in \mathbb{R}^n : ||x|| \leq 1 \} and S^n is an n-sphere. how do i show that D^n / S^{n-1} is homeomorphic to S^n? there's a hint suggesting i first of all try the n=1,2,3 cases. where X/Y= X \backslash Y \cup \{ t \} where t...

Homework Statement I want to find a bijective function from [0,1] x [0,1] -> D, where D is the closed unit disc. Homework Equations The Attempt at a Solution I have been able to find two continuous surjective functions, but neither is injective. they are...

Is it fair to think of a diffeomorphism as being a "stronger" condition then a homeomorphism? I know this is probably a dumb question, but I'm trying to teach myself some topology, and still waiting for Munkres to come in the mail. :)

I've just been reading about how complex functions can be defined on the extended complex plane. They start with rational functions as examples, and defining them at oo so they're continuous at oo in a sense. Eg, 1/z would be defined to be 0 at z = oo. I understand that given a holomorphic...

Hi, I'm trying to prove that the projective n-space is homeomorphic to identification space B^n / ~ where for x, x' \in B^n: x~x'~\Leftrightarrow~x=x' or x'=\pm x \in S^{n-1}, The way I have tried to solve this is, I introduced: {H_{+}}^{n}=\{x\in S^n | x_n \geq 0\} Then...

So for my analysis final, one of the questions was to prove the smooth version of the Hairy Ball theorem (that there is no smooth, non-vanishing function f from S^2 to itself such that for all x in S^2, f(x) is non-zero and x is tangent to f(x)) (The exam was over 2 weeks ago, so I think it's...

How can we show that the set {A in GL(n;R) | det(A)>0} is homeomorphic to the set {A in GL(n;R) | det(A)<0}?"

I posted this earlier and thought I solved it using a certain definition, which now I think is wrong, so I'm posting this again: Show that the quotient spaces R^2, R^2/D^2, R^2/I, and R^2/A are homeomorphic where D^2 is the closed ball of radius 1, centered at the origin. I is the closed...

Show the following spaces are homeomorphic: \mathbb{R}^2, \mathbb{R}^2/I, \mathbb{R}^2/D^2. Note: D^2 is the closed ball of radius 1 centered at the origin. I is the closed interval [0,1] in \mathbb{R}. THEOREM: It is enough to find a surjective, continuous map f:X\rightarrow Y to show that...

1. Let f:\mathbb{R}\rightarrow\mathbb{R} be a bijection. Prove that f is a homeomorphism iff f is a monotone function. I think I have it one way (if f is monotone, it is a homeomorphism), but I'm stuck on the other way (if f is a homeomorphism, then it is monotone). I tried to prove...

Show that tan: (-pie/2,pie/2)->R is a homeomorphism where tan = sin/cos To show that f and f^-1 are cts, it seems trivial from a sketch but how do you do it? For 1-1 tan(x) = tan(y) Need to knwo x =y tan(x) = sinx.cosx = siny/cosy = tany => sixcosy = sinycosx this gets you...

What would an explicit or analytic formula for a homeomorphism between a circle and a square be? Or a disc and [0,1] x [0,1]?

Let p,q be two prime numbers. Prove that there exists a homeomorphism of rings such that f([1]_p)=[1]_q from Z_p[X] into Z_q[X] if and only if p=q. I believe that the converse of the statement is trivial but the implication seems to be obvious? I really don't know what there really is to...

hi! would like to know what a homeomorphism means ( how do you geometrically visualize it?) AND is the symbol 8 homeomorphic to the symbol X? Why or why not? ( from whatever little i know intuitively about homeomorphisms, i think it is not...)