Let $X $ be a compact $T_0$ topological space and let $\{ A_i \}_{ i \in I }$ be a family of closed and connected subsets of $X $ and $ x\in X$ such that for each $i\in I $ there exists a closed and open (clopen) subset $ T $ of $ X $ such that $x\in T $ and $A_i\subseteq X\setminus T $. How can we find a closed and open (clopen) subset $ U $ of $ X $ such that $x\in U $ and $\cup_{ i \in I } A_i\subseteq X\setminus U $?
If it is not true , under what conditions it may be true? Note: Actually $A_i's$ are contained in different connected components.
Here is an example where it is not possible to find such a subset $U$. Let $X = \{0\} \cup_{n \in \mathbb{Z}_{\geq 2} }[\frac{1}{n},\frac{1 + n}{n^2}] \subset \mathbb{R}$.
This is Hausdorff hence $ T_{0}$ and compact since it is closed and bounded. Let $x =0$ and $A_{i} = [\frac{1}{i},\frac{1+i}{i^2}]$ for $i \geq 2$.