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Math Help - Compact Hausdorff Space

  1. #1
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    Compact Hausdorff Space

    Let X be a compact Hausdorff space and let  \{U_\alpha\}_{\alpha\in A} be an open cover of X. I want to show that there exists a finite number of continuous real-valued functions h_1,\cdots,h_m on X with the following properties:
    1. 0\leq h_j\leq 1, where 1\leq j\leq m

    2. For each 1\leq j\leq m, there is an index \alpha_j such that the closure of the set \{x\ : h_j(x)>0\} is contained in U_{\alpha_j}

    Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bram kierkels View Post
    Let X be a compact Hausdorff space and let  \{U_\alpha\}_{\alpha\in A} be an open cover of X. I want to show that there exists a finite number of continuous real-valued functions h_1,\cdots,h_m on X with the following properties:
    1. 0\leq h_j\leq 1, where 1\leq j\leq m

    2. For each 1\leq j\leq m, there is an index \alpha_j such that the closure of the set \{x\ : h_j(x)>0\} is contained in U_{\alpha_j}

    Thanks
    Got any leads. This seems like you should be able to apply Urysohn's lemma.
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