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Homework Help: 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!

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


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    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 by a moderator: 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
    For (2), you might want to try the cocountable topology.
  6. Nov 20, 2011 #5
    Thanks, I will! :-)
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