distance to a set

Let $X$ be a metric space with a metric $d$. If $A$ is a non-emptysubset of $X$ and $x\in X$, then the distance from $x$ to $A$[1] is defined as

$$d(x,A):=\underset{a\in A}{inf}d(x,a).$$

We also write $d(x,A)=d(A,x)$.

Suppose that $x,y$ are points in $X$, and $A\subset X$ is non-empty.Then we have the following triangle inequality

$d(x,A)$$=$$\underset{a\in A}{inf}d(x,a)$

$\le$$d(x,y)+\underset{a\in A}{inf}d(y,a)$

$=$$d(x,y)+d(y,A).$

If $X$ is only a pseudo-metric space, then the above definitionand triangle-inequality also hold.