Definition:Distance/Sets/Metric Spaces

Definition
Let $M = \struct {A, d}$ be a metric space.

Let $x \in A$.

Let $S, T$ be subsets of $A$.

The distance between $x$ and $S$ is defined and annotated $\ds \map d {x, S} = \inf_{y \mathop \in S} \paren {\map d {x, y} }$.

The distance between $S$ and $T$ is defined and annotated $\ds \map d {S, T} = \inf_{\substack {x \mathop \in S \\ y \mathop \in T} } \paren {\map d {x, y} }$.

Also denoted as
Some sources write $\operatorname {dist}$ instead of $d$.

Also see

 * Distance from Point to Subset is Continuous Function
 * Definition:Hausdorff Distance