Existence of disjoint open sets in a metric space.

Question

Let $X$ be a metric space, let $C$ be a closed subset of $X$, and let $x \in X\setminus C$. Prove the existence of disjoint open sets $A$ and $B$ such that $x \in A$ and $C \subset B$.

Answer 1

Draw a picture. The union of any collection open balls will yield an open set. First, choose an open ball of sufficiently small radius centered at $x$, say $B(x, \delta)$, and then to each point in $C$ you can draw an open ball centered at that point that is disjoint with $B(x, \delta)$...

Answer 2

$X\setminus C$ is open, so you can get an open ball centered in $x$ of radius $\epsilon$ $B(x,\epsilon)$ such as $B(x,\epsilon)\subset X\setminus C$

Then take:

$A=B\left(x,\frac{\epsilon}{2}\right)$

$B=\bigcup_{y\in C} B\left(y,\frac{\epsilon}{2}\right)$

It is easy to see that B is open, as union of open sets, and $A\cap B=\emptyset$

EDIT: my answer is essentialy the same to Kaj Hansen's, just that it took longer to type.