Existence of functions on a Compact hausdorff space

Question

I am supposed to use urysohn lemma in part (a) but for that i need to find closed sets in X I am not sure how to do it. I wish U1 and U2 were disjoint then it would have been easier. Help please.

Answer 1

First use that $X$ is normal to find $F_1 , F_2$ closed in $X$ so that $F_1 \subseteq U_1$ and $F_2 \subseteq U_2$ and $F_1 \cup F_2 = X$. (Note that $X\setminus U_1$ and $X\setminus U_2$ are closed and disjoint in $X$.)

Then find Urysohn functions $g_i $ to $[0,1]$ that are $1$ on $F_i$ and $0$ on $X\setminus U_i$ for $i=1,2$. Now do some scaling (both $g_i$ are already supported on $U_i$, now fix the sum property).