Do Carmo's book, chap2 Regular surfaces, definition 1.2 -- question

[Jianphys17]

On chapter over regular surfaces, In definition 1 point 2. He says that x: U → V∩S is a homeomorphisms, but U⊂ℝ^2 onto V∩S⊂ℝ^3. I am confused, how can it be so!

 

[micromass]

What's wrong with that?

 

[Jianphys17]

What's wrong with that?

The fact that U ⊆ℝ^2 but V∩S⊆ℝ^3. How can there be a homomorphism between these two spaces, the dimensions are different !

 

[micromass]

The fact that U ⊆ℝ^2 but V∩S⊆ℝ^3. How can there be a homomorphism between these two spaces, the dimensions are different !

$\mathbb{R}^3$ has dimension 3, but $V\cap S$ has dimension 2##.

 

[Jianphys17]

$\mathbb{R}^3$ has dimension 3, but $V\cap S$ has dimension 2##.

Sorry, but U⊂ℝ^2 and p∈S⊂ℝ^3, with neighborhood V⊂ℝ^3. But for homeomorphism definition, |U| ≠ |V∩S| as can be ?

 

[micromass]

$U$ and $V\cap S$ have the same number of elements and the same dimension.

 

[Jianphys17]

$U$ and $V\cap S$ have the same number of elements and the same dimension.

I had realized that dim(V∩S) = 3.
Anyway thanks for the help.

 

[micromass]

I had realized that dim(V∩S) = 3.
Anyway thanks for the help.

It doesn't have dimension 3.