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