Results 1 to 2 of 2

Math Help - analysis question

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    42

    analysis question

    Show that every open set UR is a union of at most countably many disjoint open intervals [Hint: Each point x of U lies in a unique maximal interval of U, which is the union of all intervals contained in U which contain x. Each such interval contains at least one rational number.]

    can anyone help me to approach this question ? ,,,
    thanks a lot
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,646
    Thanks
    1596
    Awards
    1
    Quote Originally Posted by jin_nzzang View Post
    Show that every open set UR is a union of at most countably many disjoint open intervals [Hint: Each point x of U lies in a unique maximal interval of U, which is the union of all intervals contained in U which contain x. Each such interval contains at least one rational number.]
    This problem is completely solved by using the hint.
    The interesting part of this is proving the hint is true.
    If x \in U define two sets L_x  = \left\{ {y:\left[ {x,y} \right) \subset U} \right\}\quad \& \quad K_x  = \left\{ {y:\left( {y,x} \right] \subset U} \right\}.
    You be able to give reasons why both sets are nonempty.

    Now look for least upper bound of L_x and greatest lower bound of K_x and with those as endpoints you have a maximal open interval containing x and is a subset of U.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Analysis question
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 4th 2011, 05:34 AM
  2. Analysis Question
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 19th 2010, 03:29 AM
  3. Analysis question
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 21st 2009, 07:16 PM
  4. Hi can anyone please help with these question on analysis.
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: April 30th 2008, 08:13 AM
  5. analysis question
    Posted in the Calculus Forum
    Replies: 9
    Last Post: November 18th 2007, 08:51 AM

Search Tags


/mathhelpforum @mathhelpforum