Results 1 to 2 of 2

Math Help - 1-manifolds

  1. #1
    Shinryu
    Guest

    1-manifolds

    Hi to all!
    I'm a new member, my name is Riccardo.
    I've a problem with 1-manifolds, i have to prove a proposition but i can't do it.

    Here is the extract of the book with notation and the proposition 1 that i have to prove:

    //
    DEFINITION. A 1-manifold is a second countable Hausdorff topological
    space X such that X can be covered by open sets each of which is
    homeomorphic either to the open interval (0,1) or the half-open
    interval [0, 1). Sets of the first type will be called O-sets, of the second
    type H-sets, of either type I-sets, and the corresponding homeomorphsms
    to these intervals will be called O-charts,H-charts, and I-charts, respectively.
    If X can be covered by O-sets it is a manifold without boundary, otherwise it
    is a manifold with boundary.

    From here on U and V will stand for I-sets in a 1-manifold and f and g will be
    associated I-charts.

    LEMMA. Suppose U∩V (set theoretic intersection) and U - V are
    nonempty and let (x_n) be a sequence in U n V converging
    to x in U - V (set theoretic difference). Then the sequence g(x_n)
    has no limit point in g(V).

    We say that U and V ouerlap if U∩V, U - V and V - U are nonempty.

    DEFINITION. An open subinterval of (0,1) is lower if it is of the form (0, b) and
    upper if it is of the form (a, 1). A subinterval which is either upper or lower is
    called outer. It is easy to see that an open interval in (0,1) is outer if and only if it
    contains a sequence with no limit point in (0,1). Similarly, in [0, 1), a subinterval is
    called upper and outer if it is of the form (a, 1). (There are, by definition, no lower
    open subintervals of [0, 1).) An open subinterval of [O,1) is outer if and only if it
    contains a sequence with no limit point in [O,1).

    PROPOSITION1. If U and V overlap and W is a component of U∩V, then f(W)
    and g(W) are outer intervals.

    Hint: Note that f(W) is a proper subinterval of f(U). Using the lemma show
    that f(W) is an open interval. Then construct an appropriate sequence in f(W)
    and use the lemma again.
    //

    Thank you in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    536
    Thanks
    10
    Suppose f|_{W}(x_n)\subset (0,1) has a limit point 0<b<1. Use the lemma on the sets U'=U\cap V-W, \ V'=W and the sequence f^{-1}\circ (f|_{W})(x_n).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ok, I know manifolds. What's next?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 15th 2011, 12:50 AM
  2. Submersion between manifolds
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: February 10th 2011, 07:04 AM
  3. orientation of manifolds
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: February 5th 2011, 06:31 AM
  4. Mobius n-manifolds within Complex n manifolds
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: January 20th 2010, 07:15 AM
  5. Manifolds
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 16th 2009, 12:44 PM

/mathhelpforum @mathhelpforum