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Homedistance to a set

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# 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.

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

54E35*no label found*

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