Homeomorphism of the cantor set to itself

[talolard]

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$$\left\{ 0,1\right\} ^{\mathbb{N}}$$so we will contruct the desired Homemorphism there.

For $$x,y\in\left\{ 0,1\right\} ^{\mathbb{N}}$$ Define $$F:_{x,y}\left\{ 0,1\right\} ^{\mathbb{N}}\rightarrow\left\{ 0,1\right\} ^{\mathbb{N}}$$ Such that the coordinate function $$F_{x,y}^{n}(z)=\begin{cases} z_{n} & z\neq x\wedge z\neq y\\ y_{n} & z=x\\ x_{n} & z=y\end{cases}.$$
The coordinate function is obviously cotinous since it is a function from a discrete space

toa discrete space and so the “product function” is also continuous. The same reasoning makes the inverse function continuous. It is easy to see the the function is 1-1 since it maps it's element to itself

except x and y which it inverses and for the same informal reasoning it is onto and so a homeomorphism.

Homework Equations

The Attempt at a Solution

 

[mighty2000]

it is important to always double check and verify any solutions or claims made by others. In this case, the solution provided is correct. The Cantor set, also known as the Cantor space, is indeed homeomorphic to the space \left\{ 0,1\right\} ^{\mathbb{N}}. Therefore, the proposed homemorphism from the Cantor set to itself is valid.

To further explain, the function F defined in the solution is a valid homemorphism because it satisfies all the necessary criteria. It is a continuous function, as it maps elements from a discrete space to a discrete space. It is also bijective, meaning it is both one-to-one and onto, which is important for a homemorphism.

In addition, the inverse function is also continuous, as it maps elements from the discrete space back to itself. This ensures that the function is well-behaved and does not create any discontinuities or distortions in the space.

Overall, the solution provided is correct and can be verified through mathematical proofs and examples. it is important to always double check and verify solutions to ensure accuracy and validity.