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Math Help - Compactness and Product Topologies

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    Super Member Aryth's Avatar
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    Compactness and Product Topologies

    So, I have a problem that I'm working on and I can't seem to figure it out. We just started product topologies, so many properties are still new or unknown to me. Here's the problem:

    Let (X,\Omega) and (Y,\Theta) be topological spaces. If A \subseteq Y is compact relative to \Theta and x \in X, show that \{x\}\times Y is compact relative to the product topology on X\times Y.

    I'm not seeing why A being compact is sufficient for the whole product to be compact... Any help would be appreciated.
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    Re: Compactness and Product Topologies

    Quote Originally Posted by Aryth View Post
    Let (X,\Omega) and (Y,\Theta) be topological spaces. If A \subseteq Y is compact relative to \Theta and x \in X, show that \{x\}\times Y is compact relative to the product topology on X\times Y.
    I'm not seeing why A being compact is sufficient for the whole product to be compact... Any help would be appreciated.
    Are you sure that you have copied the question correctly?

    Could it be show that \{x\}\times A is compact relative to the product topology on X\times Y~?.
    Thanks from Aryth
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    Super Member Aryth's Avatar
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    Re: Compactness and Product Topologies

    Quote Originally Posted by Plato View Post
    Are you sure that you have copied the question correctly?

    Could it be show that \{x\}\times A is compact relative to the product topology on X\times Y~?.
    If it is then I certainly understand it, but the question definitely says \{x\}\times Y. I'll ask my professor tomorrow if the problem as stated has a typo and prove the revision instead. Thanks for the help.
    Last edited by Aryth; September 10th 2013 at 02:07 PM.
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