1. ## 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

2. Originally Posted by bram kierkels
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.