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.
Hi, in the definition of fiber bundle there is a continuous onto map ##\pi## from the total space ##E## into the base space ##B##. Then there are local trivialization maps ##\varphi: \pi^{-1}(U) \rightarrow U \times F## where the open set ##U## in the base space is the trivializing neighborhood...
Hi,
consider the Euclidean space ##\mathbb R^8## and the projection map ##\pi## over the first 4 coordinates, i.e. ##\pi : \mathbb R^8 \rightarrow \mathbb R^4##.
I would show that the restriction of ##\pi## to the linear subspace ##A## (endowed with the subspace topology from ##\mathbb R^8##)...
Hi,
##SU(2)## group as topological space is homeomorphic to the 3-sphere ##\mathbb S^3##.
Since ##SU(2)## matrices are unitary there is a natural bijection between them and points on ##\mathbb S^3##. In order to define an homeomorphism a topology is needed on both spaces involved. ##\mathbb...
Using the QR decomposition (the complex version) I want to prove that ##SL_2(C)## is homeomorphic to the product ##SU(2) × T## where ##T## is the set of upper-triangular 2×2-complex matrices with real positive entries at the diagonal. Deduce that ##SL(2, C)## is simply-connect.
So, I can define...
Hi,
a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of...
Outline of proof:
Part I:
##1.## ##f## is a homeomorphism, so there exists a continuous inverse ##g:Y\longrightarrow X##.
##2.## ##f## is a bijection, hence there is a unique ##f(x)## in ##Y## for every ##x## in ##X##. For every ##f(x)\in Y##, the preimage under ##f## is...
let ##X=\{0,p1,p_2,...,p_n,1\}## and ##Y=\{0,p1,p_2,...,p_n,1\}## be sets equipped with the discrete topology.
for each ##q_i## in ##Y##, the inverse image ##h^{-1}(q_i)=p_i## is open in ##X## w.r.t. to the discrete topology, so h is continuous.
every element y in Y has a preimage x in X, so h...
Hi,
I know there is actually no way to set up a global coordinate chart on a 2-sphere (i.e. we cannot find a family of 2-parameter curves on a 2-sphere such that two nearby points on it have nearby coordinate values on ##\mathbb R^2## and the mapping is one-to-one).
So, from a formal...
Hi,
starting from this (old) thread
I'm a bit confused about the following: the transition function ##ϕ_{12}(b)## is defined just on the intersection ##U_1\cap U_2## and as said in that thread it actually amounts to the 'instructions' to glue together the two charts to obtain the Möbius...
Hello,
consider a full-cone (let me say a cone including bottom half, upper half and the vertex) embedded in ##E^3##. We can endow it with the topology induced by ##E^3## defining its open sets as the intersections between ##E^3## open sets (euclidean topology) and the full-cone thought itself...
Hi,
I'm a bit confused about the locally euclidean request involved in the definition of manifold (e.g. manifold ): every point in ##X## has an open neighbourhood homeomorphic to the Euclidean space ##E^n##.
As far as I know the definition of homeomorphism requires to specify a topology for...
Let ##d_1## and ##d_2## be two metrics on the same set ##X##. We say that ##d_1## and ##d_2## are equivalent if the identity map from ##(X,d_1)## to ##(X,d_2)## and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...
Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8
##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)##
As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in...
Hi,
consider an "half-cone" represented in Euclidean space ##R^3## in cartesian coordinates ##(x,y,z)## by: $$(x,y,\sqrt {x^2+y^2})$$
It does exist an homeomorphism with ##R^2## through, for instance, the projection ##p## of the half-cone on the ##R^2## plane. You can use ##p^{-1}## to get a...
Homework Statement
show that the two topological spaces are homeomorphic.
Homework Equations
Two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse between them
The Attempt at a Solution
I have tried proving that these two spaces are homeomorphic...
Hello! I just started reading something on differential geometry and I am not sure I understand the Difference between diffeomorphism and homeomorphism. I understand that the homeomorphism means deforming the topological spaces from one to another into a continuous and bijective way (like a...
Homework Statement
Let p: E-->B be a covering map. Let E be path connected and B be simply connected. Prove that p is a homeomorphism.
Homework EquationsThe Attempt at a Solution
I'm really struggling with this. Can anyone give me any insights? B is simply connected so any two paths with the...
I was trying to show that a closed interval ##[a,b]## and ##\mathbb{R}## cannot be homeomorphic. I would like to know whether this can actually be considered as a proof. It is the following:
- The closed interval ##[a,b]## can be written as ##[a,p] \cup [p,b]##, where ##a \leq p \leq...
The 2-sphere ##\mathbb{S}^2## can be expressed as the product ##\mathbb{S}^1 \times \mathbb{S}^1##
Now can we express ##\mathbb{S}^1## as ##\mathbb{S}^1 \subset (-a,a)##, where ##(-a,a)## is some open interval of ##\mathbb{R}##? If so, then (I think) ##\mathbb{S}^1## is homeomorphic to...
Hello,
I want to prove that a graph represent a manifold, for this i take the opposites edges of a vertex (edge connected between vertex connected to the current vertex) and this subgraph need to be homeomorphic for example to the 1-sphere if i want a 2 manifold. This criterion ensure that my...
Hello,
1. Homework Statement
Let be E a banach space, A a continuous automorphsim(by the banach theorem his invert is continus too.). and f a k lipshitzian fonction with $$k < \frac{1}{||A^{-1}||}$$.
Homework Equations
$$k < \frac{1}{||A^{-1}||}$$
The Attempt at a Solution
I have to show...
Hi, I'm trying to figure out how the current density for a poloidal current in toroidal solenoid is written. I found you may define a torus by an upper conical ring ##(a<r<b,\theta=\theta_1,\phi)##, a lower conical ring ##(a<r<b,\theta=\theta_2,\phi)##, an inner spherical ring...
From a topological point of view a homeomphism is the best notion of equality between topological spaces. I.e. homeomorphisms preserve properties such as Euler characteristic, connectedness, compactness etc.
I've understood it such that diffeomorphisms are the best notion of equality between...
I am independently working through the topology book called, "Introduction to Topology: Pure and Applied." I am currently in a chapter regarding manifolds. They attempt to show that a connected sum of a Torus and the Projective plane (T#P) is homeomorphic to the connected sum of a Klein Bottle...
I have one other question and I'd appreciate any insight in to. What exactly is "homeomorphism type"? I understand well what a homeomorphism is, but not what a homeomorphism type is. For example, I read about lens spaces and read things like "some lens spaces have the same homotopy type but...
It is well known that there does NOT exist a homeomorphism between R^m and R^n if m>n. My question is whether it is possible to construct a homeomorphism between R^m (as a whole) and a subset of R^n (note that we also suppose that m>n)?
Intuitively, it is impossible. Is my intuition right...
hi there
I'd like to show that the sphere
\mathbb{S}^n := \{ x \in \mathbb{R}^{n+1} : |x|=1 \} is the one-point-compactification of \mathbb{R}^n (*)
After a lot of trying I got this function:
f: \mathbb{S}^n \setminus \{(0,...,0,1)\} \rightarrow \mathbb{R}^n
(x_1,...,x_{n+1})...
I'm trying to prove the homeomorphism between the open intervals
of the real line and the open sets
of the circle with the induced topology of R^2.
Notice that the open sets of the circle is the intersection between
the open balls in R^2 and the circle itself.
Anyone can help me...
Geometrically, what is the difference between saying 'X is homotopic equivalent to Y' and 'X is homeomorphic to Y'? I know that a homeomorphism is a homotopy equivalence, but I can't seem to visualise the difference between them. It seems to me that both of these terms are about deforming spaces...
Homework Statement
Find an explicit homeomorphism from (0,1) to R.
Homework Equations
A homeomorphism from (-1,1) to R is f(x)=tan(pi*x/2).
The Attempt at a Solution
I'm horrible a modifying trig functions. Obviously, to shift by b you add b to (x) and you can change the...
Homework Statement
NxNx[0,1) is homeomorphic to [0, 1). Find an explicit homeomorphism.
(Note that N=naturals)
Homework Equations
A function f is a homeomorphism if:
(1) f is bijective
(2) f is continuous
(3) f inverse is continuous
The Attempt at a Solution
Finding a map from...
Homework Statement
(X,d1),(Y,d2) and (Z,d3) are metric spaces, Y is compact,
g(y) is a continuous function that maps Y->Z with a continuous inverse
If f(x) is a function that maps X->Y, and h(x) maps X->Z such that h(x)=g(f(x))
Show that if h is uniformly continuous, f is uniformly...
How would one go about proving ℝn is not homeomorphic to ℝm for m≠n. This isn't a homework question, I was told, but we weren't shown the proof.Is there somewhere with a proof I can see, or can someone outline it briefly?
I'm well aware and understand that homeomorphism do not need to preserve metric completeness, I'm just trying to work out a simple counterexample. I have tried searching around just for kicks, but only seem to find more complex ones. I'm wondering if the one I have works for it for sure?
On...
Is this claim true? Assume that X,Y are topological spaces, and that all closed subsets of X are compact. Then all continuous bijections f:X\to Y are homeomorphisms.
It looks true on my notebook, but I don't have a reference, and I don't trust my skills. Just checking.
Homework Statement
let f(x) = x3 and g(x) = x - 2x3. Show there is no homeomorphism h such that h(g(x)) = f(h(x))
Homework Equations
Def let J and K be intervals. the function f:J->K is a homeomorphism of J onto K if it is one to one, onto, and both f and its inverse are...
Hi, I need to know if the following statement is false or true. Given two topological spaces, X and Y, and an homeomorphism, F, between them, if bA is the boundary of the subset A of X, this implies that F(bA) is the boundary of the subset F(A) of Y?
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 feel like I got away easy with this one. Could somone let me know if I got it wrong?
Thanks
Is there a homemorphism from the Cantor set C to itself sucht hat for some x,y\in C f(x)=y
Solution. Yes
We know that the canot set is homemorphic to the space \left\{...
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...