
zero dimensional space
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 "zerodimensional" 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 0dimensionality). 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.