Results 1 to 6 of 6

Math Help - Topology and Open Set Question

  1. #1
    Banned
    Joined
    Jan 2009
    Posts
    20

    Topology and Open Set Question

    Define a topology on \Re(by listing the open sets within it) that contains the open set (0,2) and (1, 3) and that contains as few open sets as possible.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Dec 2008
    From
    Indiana
    Posts
    127
    Quote Originally Posted by r2dee6 View Post
    Define a topology on \Re(by listing the open sets within it) that contains the open set (0,2) and (1, 3) and that contains as few open sets as possible.
    You basically need to go back to the definition of a topology.

    If \mathcal{T} is a topology on the set X,
    then X and \emptyset are in \mathcal{T}

    And if there are A, B are open in \mathcal{T}, then A \cap B is open in \mathcal{T} also.

    And if there are A, B, ... are open in \mathcal{T}, then A \cup B is open in \mathcal{T} also.

    Can you take it from here?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,707
    Thanks
    1470
    I think the smallest such topology contains 8 sets- but there are several correct answers.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Dec 2008
    From
    Indiana
    Posts
    127
    Quote Originally Posted by HallsofIvy View Post
    I think the smallest such topology contains 8 sets- but there are several correct answers.
    Interesting. I have only come up with 6.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jan 2009
    Posts
    23
    It's also important to know: If each set \mathcal{A}_1,\mathcal{A}_2,\cdots,\mathcal{A}_n is open, then
    \mathcal{M}_1:=\bigcup_{k=1}^{n}\mathcal{A}_k
    \mathcal{M}_2:=\bigcap_{k=1}^{n}\mathcal{A}_k
    are open, if closed then closed. There's also a simple proof.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,707
    Thanks
    1470
    Quote Originally Posted by chabmgph View Post
    Interesting. I have only come up with 6.
    You are completely right. (I did say "I think"!) For some reason I was thinking that every we would need to add something like (-\infty, 3) and (0, \infty) to get all real numbers but we don't- they are already in R.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Compact-Open Topology
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: April 18th 2010, 05:44 PM
  2. Basic Topology, closed and open sets
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 3rd 2010, 11:43 PM
  3. Open and Closed set topology question
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 17th 2009, 12:42 PM
  4. Topology Open/Closed Sets
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 27th 2008, 06:52 PM
  5. topology - open sets vs neighborhoods
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: May 23rd 2007, 04:08 PM

Search Tags


/mathhelpforum @mathhelpforum