Solving the Homeomorphism Problem Using Real Sequences and Topology Theorems

In summary, to show that the function h is a homeomorphism, we first prove its continuity using a theorem that states if each of its components is continuous, then the function is continuous. Then, we prove that it is one-to-one and onto, and its inverse is also continuous. This is done by showing that each component of h is continuous and that for every point in the image set of h, there exists a point in the domain that maps to it. Finally, we provide an alternative proof for the continuity of h using neighborhoods.
  • #1
radou
Homework Helper
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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 homeomorphism.

Homework Equations



I used a theorem which states that if f : A --> ∏Xj is given by the equation f(a) = (fj(a)) (j is from some indexing set J), where fj : A --> Xj, for each j, then f is continuous if and only if fj is continuous, for each j.

I'm not sure if I can use this theorem here, since there's no information about what the set A is.

The Attempt at a Solution



First I tried to prove that h is continuous using the theorem up there.

h(x) can be represented with (h1(x), h2(x), ...), where hj(x) = aj∏j(x) + bj. Here ∏j(x) denotes the projection mapping onto the j-th coordinate of x. Since hj(x) is continuous for every j (the projection mapping is continuous, addition and multiplication are, too), we conclude that h is continuous.

Proving one to one and onto is easy:

Let h(x) = h(y), then for every j, ajxj + bj = ajyj + bj, hence xj = yj, for every j. Let c = (c1, c2, ...) be in the image set of h. Then, for every j, (cj - bj)/aj = xj, and hence (x1, x2, ...) maps to c.

The j-th component of the inverse of h is given with (hj(x) - bj)/aj, and the inverse is continuous too, since it's continuous for every index j.

All this is well defined, since aj > 0, for every j.

Hence, h is a homeomorphism from R∞ to R∞.
 
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  • #2
Here's an alternative proof for continuity, which seems even easier.

Let x be a point in Rω, and Uω a neighborhood of h(x). Every component of h(x) lies in a neighborhood <a, b>, i.e. a < aj xj + bj < b. It follows that <(a-bj)/aj, (b-bj)/aj> is a neighborhood of xj whose image is in <a, b>. The product of all such neighborhoods in Rω is a neighborhood of x whose image under h lies in Uω.
 

1. What is the Homeomorphism Problem?

The Homeomorphism Problem is a mathematical concept that deals with determining whether two topological spaces are homeomorphic, or in simpler terms, whether they can be transformed into each other without tearing or gluing. It is a fundamental problem in topology and has applications in various fields such as physics, computer science, and biology.

2. How is the Homeomorphism Problem solved?

The Homeomorphism Problem is solved by using various techniques and tools from topology, such as continuous functions, open and closed sets, and deformation retracts. The most common method is to construct a homeomorphism, which is a bijective function between two spaces that preserves the topological structure.

3. Why is the Homeomorphism Problem important?

The Homeomorphism Problem is important because it helps us understand the properties and relationships between different topological spaces. It also allows us to classify and differentiate between spaces, which is crucial in many areas of mathematics and science.

4. What are some real-world applications of the Homeomorphism Problem?

The Homeomorphism Problem has applications in various fields, such as computer graphics, where it is used to create realistic 3D models and animations. It is also used in physics to study the behavior of fluids and solids, and in biology to understand the shape and structure of biomolecules.

5. Are there any unsolved aspects of the Homeomorphism Problem?

Yes, there are still some unsolved aspects of the Homeomorphism Problem, such as the Poincaré Conjecture and the Schoenflies Conjecture. These conjectures deal with the classification of 3-dimensional manifolds and have been proven to be true in specific cases, but a general proof is still unknown.

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