Distance in Metric Spaces

Definition: Distance between Points

Let $(M,d)$ be a metric space.

The distance between two points $x,y\in M$ is

$$d(x,y)$$

Definition: Distance between a Point and a Subset

Let $(M,d)$ be a metric space.

The distance between a point $x\in M$ and a subset $S\subseteq M$ is the infimum of the ^distance-between-points between $x$ and the points of $S$.

$$\inf \{d(x,s)\mid s\in S\}$$

NOTATION

$$d(x,S)$$