\begin{definition}[Definition:Point-to-Set Distance]
Let $\XX$ be a normed space.
The '''point-to-set distance''' on $\XX$ is a mapping:
:$d: \XX \times \powerset \XX \to \hointr 0 \to$
where $\powerset \XX$ is the power set of $\XX$.
This mapping is defined as:
:$\map d {x, C} := \inf \set {\norm {x - y}: y \in C}$
with the convention that for any $x \in \XX$:
:$\map d {x, \O} = +\infty$
or, to be more mathematically rigorous:
:$\forall C \in \powerset \XX \setminus \set \O: \map d {x, \O} > \map d {x, C}$
Category:Definitions/Normed Spaces
\end{definition}