What is Homeomorphism: Definition and 71 Discussions
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 1895.Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle.
An often-repeated mathematical joke is that topologists cannot tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in the cup's handle.
1. (1) (a) Let X be a topological space. Prove that the set Homeo(X) of home-
omorphisms f : X → X becomes a group when endowed with the binary operation f ◦ g.
(b) Let G be a subgroup of Homeo(X). Prove that the relation ‘xRG y ⇔ ∃g ∈ G such
that g(x) = y ’ is an equivalence relation.
(c)...
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...
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...
Homework Statement
Prove that D^n/S^{n-1} is homeomorphic to S^n . (Hint - try the cases n = 1,2,3 first).
Homework Equations
X/Y is defined as the union of the complement X\Y with one point. I showed in a previous section that the equivalence relation x~x' if x and x' are in Y gives...
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...
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...
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...)