1. ## 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

2. Originally Posted by bram kierkels
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.