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Thread: Compact Hausdorff Space

  1. #1
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    Compact Hausdorff Space

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

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

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

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

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