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

• Results about the distance function can be found here.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Definition $5.8$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: Exercise $4.3: 5$
• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) $\text I.2.4$