What is the definition of distance between a point and a set of points?

[kakarukeys]

for simplicity, consider the real number space
the distance between two points x, y (two reals) is $$|x - y|$$
Is there a definition of distance between a point x and a subset of R, such as an interval (a, b)?

If there isn't any, how would you define it, such that there are some meaningful constructions?

 

[whozum]

Maybe the distance between a point and the locus {}

 

[master_coda]

The standard definition of the distance between a point x and a set S is:

$$d(x,S)=\inf\lbrace d(x,y)\colon y\in S\rbrace$$

where d is your distance function.

Basically, the distance between a point and a set is the minimum distance between the point and every element in the set. That's not completely correct, since there may not be a minimum (which is why we use "inf" and not "min"), but that's the basic idea.

 

[kakarukeys]

according to the definition, the distance between 1 and (2,3) is the inf which is |1 - 2| = 1?
If the set is an empty set, what is the distance?

 

[HallsofIvy]

The DEFINITION of "the distance between the point p and the set of points A" is
"The greatest lower bound of all distances from p to each point in A"

That is guaranteed (by the greatest lower bound property) to exist as long as A is NOT EMPTY.

The distance from a point to the empty set is not defined.