1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Topology, counter examples needed.

  1. Nov 19, 2011 #1
    1. The problem statement, all variables and given/known data

    I need two counter examples, that show the following two theorems don't/B] hold:
    Let X be a topological space.

    1. If from the closeness of any subset A in X follows compactness of A, then X is compact.
    2. If from the compactness of a subset A in X follows closeness of A, then X is housdorff.

    I proved the *opposite* theormes which do hold, but I cannot seem to find counter examples.

    That means, I need to find a non-compact space, in which every closed subset is compact, and a non-housdorff space, in which every compact subset in closed.

    2. Relevant equations

    Are there any equations in Topology?

    3. The attempt at a solution

    I just tried to take for "1" spaces I know that are not compact, but then couldn't find a space in which every closed subset is a compact one...
    In "2" I thought of non-hausdorff spaces I know, but couldn't directly see whether indeed every compact set is closed.

    I'd really appreciate your help!

    Thanks,
    Tomer.
     
    Last edited: Nov 19, 2011
  2. jcsd
  3. Nov 20, 2011 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You aren't going to be able to find a counter example to (1) because it is true. Every space is closed in itself. If it is true that every closed subset of A is compact, then A itself is compact because A is a closed subset of itself.
     
    Last edited: Nov 20, 2011
  4. Nov 20, 2011 #3
    Thanks a lot for the reply.
    I see you're point :-) That is strange - in the task we were to prove the next theorem (part a):
    If X is compact, it follows from the closeness of A that A is compact.

    Then we need to give counter examples to show why the opposite doesn't hold. I translated the "opposite claim" correctly, right?

    But I definitely agree with what you just said :-)
     
  5. Nov 20, 2011 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    For (2), you might want to try the cocountable topology.
     
  6. Nov 20, 2011 #5
    Thanks, I will! :-)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Topology, counter examples needed.
Loading...