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Math Help - Homeomorphic spaces

  1. #1
    MHF Contributor Drexel28's Avatar
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    Homeomorphic spaces

    I came up with this nice little theorem, can anyone verify or deny it?

    I can show my (very very long...need like six lemmas) proof, but it goes like this:
    Theorem:: Let \left\{X_j\right\}_{j\in\mathcal{J}} and \left\{Y_j\right\}_{j\in\mathcal{J}} be two collections of topological spaces such that X_j\approx Y_j for all j\in\mathcal{J} ( \approx means homeomorphic). Then, \prod_{j\in\mathcal{J}}X_j\approx\prod_{j\in\mathc  al{J}}Y_j under the product topology.


    Any comments would be nice.
    Last edited by Drexel28; February 20th 2010 at 08:25 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Drexel28 View Post
    I came up with this nice little theorem, can anyone verify or deny it?

    I can show my (very very long...need like six lemmas) proof, but it goes like this:
    Theorem:: Let \left\{X_j\right\}_{j\in\mathcal{J}} and \left\{Y_j\right\}_{j\in\mathcal{J}} be two collections of topological spaces such that X_j\approx Y_j for all j\in\mathcal{J} ( \approx means homeomorphic). Then, \prod_{j\in\mathcal{J}}X_j\approx\prod_{j\in\mathc  al{J}}Y_j under the product topology.


    Any comments would be nice.
    It turns out this is correct.

    If anyone wants to see a proof, take a loot at my blog at the bottom of the page.
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