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Open subsets and manifolds

  1. Nov 1, 2013 #1
    Hello,

    I was wondering if it is true that any open subset Ω in ℝn, to which we can associate an atlas with some coordinate charts, is always a manifold of dimension n (the same dimension of the parent space).
    Or alternatively, is it possible to find a subset of ℝn that is open, but it is a manifold of dimension lower than n?

    In ℝ3 I cannot think of any open subset that would be a curve or a surface. So it would seem that in this case open subset implies manifold of dimension 3 (provided we can find atlas and charts to cover it).

    Does this hold in general?

    Thanks.
     
    Last edited: Nov 1, 2013
  2. jcsd
  3. Nov 1, 2013 #2

    lavinia

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    By definition,in an open subset of Euclidean space every point has a neighborhood that is an open ball. The coordinate transformations can be taken to be the identity map.
     
  4. Nov 1, 2013 #3

    HallsofIvy

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    The definition of "open set" requires that, for every point, there exist a "neighborhood" of that point contained in the set. "Neighborhoods" in an n-dimensional manifold are themselves n-dimensional so the answer to your question is "no". An open set in an n-dimensional manifold must be n-dimensional.
     
  5. Nov 1, 2013 #4

    lavinia

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    The concept of dimension is well defined. That is: an open subset of Euclidean space can not be open in either a higher of lower dimensional Euclidean space.
     
    Last edited: Nov 1, 2013
  6. Nov 1, 2013 #5

    WWGD

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