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Math Help - example for a non compact space?

  1. #1
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    example for a non compact space?

    hi
    I need to find examples for these two topological spaces:
    a. A space X such that every sequence has a convergent subsequence, but X is not compact.
    b. Space X. every infinite set S \subseteq X has an accumulation point and X is not compact.

    thanks
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  2. #2
    Senior Member roninpro's Avatar
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    You probably need to look at topological spaces that are not metrizable. If X were a metric space, then condition (a) would actually be equivalent to compactness.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by aharonidan View Post

    b. Space X. every infinite set S \subseteq X has an accumulation point and X is not compact.
    Consider X=[0,1]^\Omega where \#(\Omega)>\aleph_0 with the product topology and define


    \mathcal{X}=\left\{(x_\omega)\in X:x_\omega=0\text{ for all but countably many }\omega\in\Omega\right\}


    Let Y be an infinite subset of \mathcal{X}. That space does the trick. Try proving it.
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