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

  1. Jul 19, 2016 #1
    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. jcsd
  3. Jul 19, 2016 #2
    What's wrong with that?
     
  4. Jul 20, 2016 #3
    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 !! :confused:
     
  5. Jul 20, 2016 #4
    ##\mathbb{R}^3## has dimension 3, but ##V\cap S## has dimension 2##.
     
  6. Jul 20, 2016 #5
    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 ?
     
  7. Jul 20, 2016 #6
    ##U## and ##V\cap S## have the same number of elements and the same dimension.
     
  8. Jul 20, 2016 #7
    I had realized that dim(V∩S) = 3.
    Anyway thanks for the help.
     
  9. Jul 20, 2016 #8
    It doesn't have dimension 3.
     
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