Distance from a point to a set

Definition

Let $(X, d)$ be a metric space, let $A$ be a nonempty subset of $X$ and let $x \in X$. The distance from $x$ to $A$ is the real number $$d(x, A) = \inf \{ d(x, a) : a \in A \}.$$

Remarks

The infimum exists, because the set is nonempty and bounded below by $0$. If $x \in A$, then $d(x, A) = 0$. The converse is false: in $\mathbb{R}$, $d(0, (0, 1)) = 0$ but $0 \notin (0, 1)$. In general $d(x, A) = 0$ if and only if $x$ is in the closure of $A$. For all $x, y \in X$, $|d(x, A) - d(y, A)| \le d(x, y)$. Thus $x \mapsto d(x, A)$ is Lipschitz with constant $1$.

Explanations

L1 human

The distance from a point to a set is the shortest distance from the point to the set, when a shortest distance exists. For example, the distance from $3$ to the interval $[0, 1]$ is $2$.