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

1. Jul 19, 2016

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!!

2. Jul 19, 2016

micromass

What's wrong with that?

3. Jul 20, 2016

Jianphys17

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 !!

4. Jul 20, 2016

$\mathbb{R}^3$ has dimension 3, but $V\cap S$ has dimension 2$. 5. Jul 20, 2016 Jianphys17 Sorry, but U⊂ℝ^2 and p∈S⊂ℝ^3, with neighborhood V⊂ℝ^3. But for homeomorphism definition, |U| ≠ |V∩S| as can be ? 6. Jul 20, 2016 micromass$U$and$V\cap S## have the same number of elements and the same dimension.

7. Jul 20, 2016

Jianphys17

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

8. Jul 20, 2016

micromass

It doesn't have dimension 3.