# 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 DistanceToASet 2013-03-22 13:38:37 2013-03-22 13:38:37 bbukh (348) bbukh (348) 4 bbukh (348) Definition msc 54E35