distance to a set

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


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

Suppose that x,y are points in X, and AX is non-empty. Then we have the following triangle inequalityMathworldMathworldPlanetmath

d(x,A) = infaAd(x,a)
= d(x,y)+d(y,A).

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


  • 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