Results 1 to 8 of 8

Math Help - Countable Topology

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Countable Topology

    X is a set and \tau_c is a collection of all the subsets U of X such that X-U either is countable or X.

    X-\bigcup_{\alpha\in A}U_{\alpha}=\bigcap_{\alpha\in A}(X-U_{\alpha}). (DeMorgan's Law--I understand this)

    Now, it says the complement of \bigcup_{\alpha\in A}U_{\alpha} is a subset of countable sets.

    I don't get this. Can someone explain what is going on?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Countable Topology

    Quote Originally Posted by dwsmith View Post
    X is a set and \tau_c is a collection of all the subsets U of X such that X-U either is countable or X.
    X-\bigcup_{\alpha\in A}U_{\alpha}=\bigcap_{\alpha\in A}(X-U_{\alpha}). (DeMorgan's Law--I understand this)
    Now, it says the complement of \bigcup_{\alpha\in A}U_{\alpha} is a subset of countable sets.
    Disclaimer: I am not at all sure that I follow this question.
    \left( {\bigcup {U_\alpha  } } \right)^c  = \bigcap {\left( {U_\alpha  } \right)^c  = \bigcap {\left( {X\backslash U_\alpha  } \right)} }
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Countable Topology

    How do we know or how can it be shown that the intersection is countable?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2010
    From
    Clarksville, ARk
    Posts
    398

    Re: Countable Complement Topology

    Quote Originally Posted by dwsmith View Post
    X is a set and \tau_c is a collection of all the subsets U of X such that X-U either is countable or X.

    X-\bigcup_{\alpha\in A}U_{\alpha}=\bigcap_{\alpha\in A}(X-U_{\alpha}). (DeMorgan's Law--I understand this)

    Now, it says the complement of \bigcup_{\alpha\in A}U_{\alpha} is a subset of countable sets.

    I don't get this. Can someone explain what is going on?

    Thanks.
    It looks like you're working with the Countable Complement Topology.

    Is this part of an exercise in which you show that the collection \tau_c is a topology on X by showing that an arbitrary union of sets that are in \tau_c, is itself in \tau_c?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Countable Topology

    Quote Originally Posted by dwsmith View Post
    How do we know or how can it be shown that the intersection is countable?
    Any subset of a countable set is countable.
    An intersection of a collection of sets is a subset of each set in the collection.
    Last edited by Plato; September 1st 2011 at 05:25 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Countable Topology

    Quote Originally Posted by Plato View Post
    Any subset of a countable set is countable.
    An intersection of a collection of sets is a subset of each set in the collection.
    Why is the set countable instead of not countable (that is what I don't understand)?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Countable Topology

    Quote Originally Posted by dwsmith View Post
    Why is the set countable instead of not countable (that is what I don't understand)?
    Look carefully at the definition.
    For each \alpha the set X\setminus U_{\alpha} is either X or is countable.
    Thus each set in that intersection is either X or is countable.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Countable Topology

    Quote Originally Posted by Plato View Post
    Look carefully at the definition.
    For each \alpha the set X\setminus U_{\alpha} is either X or is countable.
    Thus each set in that intersection is either X or is countable.
    Ok thanks. I wasn't sure I could assume that from the definition. I thought I would have to show it was countable and fit into the definition to be true.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: January 9th 2013, 06:14 AM
  2. Replies: 2
    Last Post: November 11th 2010, 04:56 AM
  3. [SOLVED] Countable union of closed sets/countable interesection of open sets
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 8th 2010, 01:59 PM
  4. Replies: 1
    Last Post: February 9th 2010, 01:51 PM
  5. Replies: 11
    Last Post: October 11th 2008, 06:49 PM

/mathhelpforum @mathhelpforum