distance to a set

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

d(x,A):=\inf_{a\in A}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

$\displaystyle d(x,A)$	$\displaystyle=$	$\displaystyle\inf_{a\in A}d(x,a)$
	$\displaystyle\leq$	$\displaystyle d(x,y)+\inf_{a\in A}d(y,a)$
	$\displaystyle=$	$\displaystyle d(x,y)+d(y,A).$

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

References

1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.

Title	distance to a set
Canonical name	DistanceToASet
Date of creation	2013-03-22 13:38:37
Last modified on	2013-03-22 13:38:37
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	4
Author	bbukh (348)
Entry type	Definition
Classification	msc 54E35