Results 1 to 7 of 7

Math Help - Sets in Metric spaces question.

  1. #1
    Member
    Joined
    May 2008
    Posts
    86

    Sets in Metric spaces question.

    Hello, thank you very much for reading.

    I'm totally stuck on the following statement, which I need to prove:

    Let {Ki} be the group of compact sets, which also satisfy the fact that every *final* intersection of them is NOT empty.
    I need to prove that any intersection of the Ki's is not empty (non-final intersections as well).

    I'm trying to play with the fact that if a set is compact, it is partial to a final unification of open sets.... But I can't see how to prove it :-\

    Replies are most welcomed and wanted!!!

    thanks,
    Tomer.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1
    What does *final* intersection mean?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    Posts
    86
    I'm sorry, Enlish isn't my native tounge.
    By final I mean... a finite number of sets in the intersection (as opposed to infinite).

    Actually, I have already submitted my assignment, with this question partially solved, and afterwords asked my proffesor to explain the answer for me, so it's not relavent anymore.
    Thank you very much, regardless!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    This is called the finite intersection property for compact sets.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    May 2008
    Posts
    86
    Wow, a very elegant proof! Too bad it came a bit late
    Thanks!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1
    Quote Originally Posted by aurora View Post
    Wow, a very elegant proof!
    BUT! Be careful. You did not state in the original question that the entire space is compact. Now that may be a language problem as well. From the wording of your post is seems that we have a collection (not a group) of compact subsets and the collection has the FIP.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Plato View Post
    BUT! Be careful. You did not state in the original question that the entire space is compact. Now that may be a language problem as well. From the wording of your post is seems that we have a collection (not a group) of compact subsets and the collection has the FIP.
    The FIP theorem works in any topological space (compact or not). It says that if a collection of compact subsets has the property that the intersection of any finite subcollection is nonempty then the intersection of the entire collection is nonempty.

    To prove it, you can replace the ambient space by one of the sets in the collection. In that way, you can assume that you are working in a compact space.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Any metric spaces can be viewed as a subset of normed spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: December 15th 2011, 04:00 PM
  2. Real Analysis and Metric Spaces question
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: December 4th 2011, 02:15 PM
  3. Metric spaces, open sets, and closed sets
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 16th 2011, 06:17 PM
  4. Metric spaces and Sets...
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: May 26th 2008, 05:47 PM
  5. question on metric spaces
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 11th 2008, 02:26 PM

Search Tags


/mathhelpforum @mathhelpforum