When are the distance between points and sets well-defined? Let $G$ be an open subset of $\mathbb C$. I would like to prove that this set $\{z\in G; d(z,\mathbb C-G)\ge 1/n\}$, where $n\in \mathbb R$, is well-defined. In another words, I would like to know if this distance is always well-defined for an open set $G\subset \mathbb C$.
 A: Conventionally, in a metric space $(X,d)$ (in your case it's just the complex plane with Euclidean metric), the distance between any two nonempty sets $A,B$ can be defined as 
$$d(A,B):=\inf_{x\in A,y\in B} d(x,y).$$
(The infimum is of course defined because of the non-emptiness.)
[EDIT: Also note that the inf is not necessarily attained in general. A quick example is $A=\{1/n\},B=\{0\}$. Hence, even if two sets are zero-distanced, it doesn't imply that they intersect. 
However, if one set is closed and the other one is compact, then zero distance is indeed equivalent to intersection. This follows from the facts that:
1). $d(z,B)$ when regarded as a function of $z$ is continuous. 
2). Continuous functions on compact sets attain their minimum values. 
3). Limit points of closed sets don't overflow. ]
If $A=\{x\}$,  containing exactly one point, then we conventionally write $d(x,B)$ for $d(A,B)$. 
If, furthermore, $B=\{y\}$ is also a single-point set, then we will write $d(x,y)$, which you should check is compatible with the original definition of the metric $d$. 
