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I want to prove the following:

Let $X$ be second countable zero-dimensional space. If $A,B \subseteq X$ are disjoint closed sets, there exist is a clopen set $C$ such that $A\subseteq C$ and $B\cap C = \emptyset $.

(A topological space $X$ is zero-dimensional if it is Hausdorff and has basis consisting of clopen sets.)

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6.2.7. THEOREM. Every zero-dimensional Lindelöf space is strongly zero-dimensional.

PROOF. It suffices to show that for every pair $A$, $B$ of disjoint closed subset of a zero-dimensional Lindelöf space $X$ there exists an open-and-closed set $U\subset X$ such that $A\subset U \subset X\setminus B$. For every $x\in X$ choose an open-and-closed set $W_x\subset X$ which contains $x$ and satisfies $$A\cap W_x = \emptyset \qquad\text{or}\qquad B\cap W_x=\emptyset.$$ Let $\{W_{x_i}\}_{i=1}^\infty$ be a countable subcover of the cover $\{W_x\}_{x\in X}$ of the space $X$. The sets $$U_i:=W_{x_i}\setminus \bigcup_{j<i} W_{x_i},\qquad\text{where }i=1,2,\dots$$ are open-and-closed and pairwise disjoint, and the family $\{U_i\}_{i=1}^\infty$ is a cover of the space $X$. The set $U=\bigcup\{U_i : A\cap U_i \ne \emptyset\}$ has the required properties.

Source: Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

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