# 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.

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1. ### I Fiber bundle homeomorphism with the fiber

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...
2. ### I Is the Projection Restriction to a Linear Subspace a Homeomorphism?

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##)...
3. ### I ##SU(2)## homeomorphic with ##\mathbb S^3##

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...
4. ### I Proving SL_2(C) Homeomorphic to SU(2)xT & Simple Connectedness

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...
5. ### I Clarification about submanifold definition in ##\mathbb R^2##

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...
6. ### Prove that f is a homeomorphism iff g is continuous, fg=1 and gf=1

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...
7. ### Prove that a function from [0,1] to [0,1] is a homeomorphism

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...
8. ### I Global coordinate chart on a 2-sphere

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...
9. ### I How Do Transition Functions Define a Möbius Strip in Fiber Bundles?

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...
10. ### I Is a Full-Cone in E^3 a Topological Manifold?

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...
11. ### I About the definition of a Manifold

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...
12. ### A Structure preserved by strong equivalence of metrics?

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...
13. ### I Metric Homeomorphism: Isometry Equivalence?

Is every homeomorphism between a metric space X and a topological Y equivalent to an isometry? I think it is, but I need to confirm.
14. ### I Injective immersion that is not a smooth embedding

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...
15. ### I Differential structure on a half-cone

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...
16. ### Finding homeomorphism between topological spaces

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...
17. ### I Difference between diffeomorphism and homeomorphism

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...
18. ### Proving conditions for a covering map to be a homeomorphism

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...
19. ### I Proving non homeomorphism between a closed interval & ##\mathbb{R}##

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...
20. ### I Boundary and homeomorphism

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...
21. ### A Is Your Graph Homeomorphic to a Sphere?

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...
22. ### Homeomorphism in a Banach space

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...
23. ### Poloidal current in toroidal solenoid

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...
24. ### What Properties Are Preserved by Diffeomorphisms?

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...
25. ### MHB Homeomorphism between a cylinder and a plane?

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
26. ### Homeomorphism of Unit Circle and XxX Product Space

Is there atopological space X such that XxX (the product space) is homeomorphic to the unit circle in the plane
27. ### Homeomorphism through cutting and pasting of manifolds

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...
28. ### MHB Homeomorphism Between Q and Unit Sphere in R^3

Find a homeomorphism between Q={(x,y,z):$x^2+y^6+z^{10}=1$} and the unit sphere in R^3
29. ### What is Homeomorphism Type? Definition & Examples

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...
30. ### A fundamental question on homeomorphism

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...
31. ### Finding inverse for a homeomorphism on the sphere (compactification)

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})...
32. ### Homeomorphism between the open sets of the circle and the open sets of real line

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...
33. ### Geometric difference between a homotopy equivalance and a homeomorphism

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...
34. ### How Can We Map the Open Interval (0,1) to the Real Line R Using a Homeomorphism?

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...
35. ### (Topology Problem) Finding an interesting homeomorphism

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...
36. ### Show that a homeomorphism preserves uniform continuity

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...
37. ### How to Prove ℝn is Not Homeomorphic to ℝm for m≠n?

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?
38. ### Example of a Homeomorphism f: Q -> Q not order preseving or order-reversing

Any ideas? I really can't think of any myself, as I'm quite the amatuer at topology.
39. ### Is this a homeomorphism that does not preserve metric completeness?

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...
40. ### From continuity to homeomorphism, compactness in domain

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.
41. ### Prove there is no homeomorphism that makes two functions conjugate

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...
42. ### Does an Homeomorphism Preserve Boundary Correspondence in Topological Spaces?

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?
43. ### Proving Homeomorphism between Topological Spaces (X,T) and (delta,U_delta)

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...
44. ### Homeomorphism of the cantor set to itself

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\{...
45. ### Homeomorphism and diffeomorphism

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...
46. ### Homeomorphism classes of compact 3-manifolds

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...
47. ### Is the Closed Unit Square Homeomorphic to the Closed Unit Disc?

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.
48. ### Solving the Homeomorphism Problem Using Real Sequences and Topology Theorems

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...
49. ### Monotonic and Continuous function is homeomorphism

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)...
50. ### Definition of a homeomorphism between topological spaces

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...