Page 2 of 2 FirstFirst 12
Results 16 to 22 of 22

Math Help - Open Sets on the Real Line

  1. #16
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98
    Quote Originally Posted by Plato View Post
    There are only three basic forms of connected open subsets of real numbers: (-\infty,a),~(a,b),~\&~(a,\infty).
    The theorem means: Any open set of real numbers can be written as a countable union of pairwise disjoint basic open connected sets.

    The set [1,\infty )\backslash \mathbb{Z}^ + = \bigcup\limits_{n = 1}^\infty {\left( {n,n + 1} \right)} therefore  [1,\infty )\backslash \mathbb{Z}^ + is an open set.

    The set (2,3) is a basic open set by definition.
    If you delete the integers from [1,infinity) you are left with the sets (0,1), (1,2), (2,3) ..... and you are left with the trivial observation that the union of a collection of disjoint open sets is the union of the same collection of disjoint open sets.

    Personally, I vote for Kolmogorov. I think the whole point is that an open set can be expressed as the union of disjoint sets not necessarily open (on the real line). Even Taylor in his theorem does not refer specifically to E1, E2... as open sets even though conditions (i)-(iii) do. That's what makes his Theorem so maddening.
    Last edited by Hartlw; January 31st 2011 at 12:54 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #17
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by Hartlw View Post
    Personally, I vote for Kolmogorov. I think the whole point is that an open set can be expressed as the union of disjoint sets (on the real line).
    What is your point there?
    I simply gave an example. I have used that book as a textbook also.
    Follow Math Help Forum on Facebook and Google+

  3. #18
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98
    Quote Originally Posted by Plato View Post
    What is your point there?
    I simply gave an example. I have used that book as a textbook also.

    Sorry, I added an edit after your response:

    "I think the whole point is that an open set can be expressed as the union of disjoint sets not necessarily open (on the real line). Even Taylor in his theorem does not refer specifically to E1, E2... as open sets even though conditions (i)-(iii) do. That's what makes his Theorem so maddening."

    OK, to be even more specific, I think Taylor is wrong and youi don't. That's probably as good a place to end it as any.
    Follow Math Help Forum on Facebook and Google+

  4. #19
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1
    Quote Originally Posted by Hartlw View Post
    "THE STRUCTURE OF OPEN SETS IN R
    Theorem 2-3 IV. If E is an open set in R such that E\neq 0, and E\neq R, E can be expressed as a countable (possibly finite) union of disjoint sets E1, E2,..., where each En is a non-empty open set of one of the following types:
    (i) a finite open interval {x : a<x<b},
    (ii) a left-semiinfinite open interval {x : x<b}, where b\in R,
    (iii) a right-semiinfinite open interval {x : a<x}, where a\in R."
    Quote Originally Posted by Hartlw View Post
    "I think the whole point is that an open set can be expressed as the union of disjoint sets not necessarily open (on the real line). Even Taylor in his theorem does not refer specifically to E1, E2... as open sets even though conditions (i)-(iii) do.
    OK, to be even more specific, I think Taylor is wrong and youi don't. That's probably as good a place to end it as any.
    How can you say Taylor does not say the E_n are open? You posted it yourself that he does.
    Follow Math Help Forum on Facebook and Google+

  5. #20
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98
    Quote Originally Posted by Plato View Post
    How can you say Taylor does not say the E_n are open? You posted it yourself that he does.
    You are right, I missed that. Technically, I copied it from Taylors book. OK, I don't understand Taylor's version.

    For reference I include Kolmogorov's version again which makes sense to me (note reference below takes you directly to the theorem, you don't have to hunt for it):

    http://books.google.com/books?id=OyW...20line&f=false
    Last edited by Hartlw; February 1st 2011 at 04:56 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #21
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98
    Shilov: "Every open set on the real line is a finite or countable union of nonintersecting open intervals."

    Rudin: "Every open set in R1 is the union of an at most countable collection of disjoint segments."

    Its still a tie.

    By the way, you can get almost any used book cheap at abe.com (hope its OK to mention this. If not, say so and I will edit it out).

    EDIT: Just as a reminder: (0,1) and [1,2) are disjoint (non-intersecting).
    Last edited by Hartlw; February 1st 2011 at 07:38 AM.
    Follow Math Help Forum on Facebook and Google+

  7. #22
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98
    Shilov (clearly) and Taylor (obtuseley) state:
    Every open set S on the real line is a countable union of nonintersecting open intervals.
    Example:
    S= (1,3)\cup{(3,8)}\cup(9,12)}
    period.

    Kolmogorov and Rudin state:
    Every open set S on the real line is (can be expressed as) a countable union of nonintersecting intervals.
    Example:
    S= (1,3)\cup{(3,5)}\cup{[5,8)}\cup(9,12)}
    or
    S= (1,2)\cup{[2,3)}\cup{(3,8)}\cup{(9,11]}\cup{(11,12)}
    or.......

    The examples make everything clear. Shilov's proof became crystal clear once I understood what the theorem was saying. Taylor and Klomogorov's proofs were'nt clear, and Rudin's was an excercise.

    What confused me all along was that I thiought Taylor was saying, for example,
    If S=(1,3) it can be expressed as (1,2)\cup{(2,3)}
    Last edited by Hartlw; February 2nd 2011 at 04:29 AM.
    Follow Math Help Forum on Facebook and Google+

Page 2 of 2 FirstFirst 12

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 30th 2010, 01:50 PM
  2. Measures on Real Line and open sets.
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: October 7th 2009, 05:01 PM
  3. Real Analysis - Open/Closed Sets
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 8th 2008, 05:39 PM
  4. Real Analysis - Open/Closed Sets
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 8th 2008, 04:32 PM
  5. Real Analysis: Open and Closed Sets
    Posted in the Calculus Forum
    Replies: 13
    Last Post: October 9th 2008, 09:32 PM

Search Tags


/mathhelpforum @mathhelpforum