Let $\displaystyle X$ be a secound countable, zero dimensional space.

If $\displaystyle A,B \subset X$ are disjoint closed sets, there is a clopen set $\displaystyle C$ separating $\displaystyle A$ and $\displaystyle B$, i.e.

$\displaystyle A\subset C$ and $\displaystyle B \cap C = \emptyset$ .

Note that a "zero-dimensional" space is by definition a Hausdorff space and has a basis of clopen sets.

Does anyone have an idea how to prove this?

What i have so far is that if X fulfills the conditions then it is also metrizable, according to the Urysohn metrization theorem (It is second countable, Hausdorff and regular, which follows from its 0-dimensionality). Finding an open set that separates A and B is no problem. But i don't know how prove the existence of an open set that is also closed, i.e. "clopen", which seperates A and B.