# Is a set of points equally distant from one point (sphere) equidistant?

What is the mathematical definition of equidistant ?

Here I found:

If the distance of each object of a set of objects to the point is same, then the point is called equidistant.

The term is usually used in cases where "the set of objects" refers to more than one object. Can it be used also in case of one object only? Or is there another term for that? I am especially interested in the case of one point in 3 dimensional space (sphere).

• Well if you compare one point to a single point only, then equidistant doesn't really apply. equidistant ("equally distant") is a comparison, and you can't compare a single thing (to what would you compare it to?) – Lisa Sep 23 '13 at 13:30
• If you want to compare your point to a sphere, then your point would be equidistant to the set of points in your sphere. – Lisa Sep 23 '13 at 13:32
• @Lisa Good point, I guess, I was getting the idea of equidistant somehow wrong. – MasterPJ Sep 23 '13 at 16:59

Given a single point $p$ in a metric space $(M,d)$, you can definitely define a set of points $S$ such that $d(s,p)=d(s',p)$ for all $(s,s')\in S\times S$. One might call $S$ a segment of the sphere of radius $d(s,p)$ centered at $p$.
In this case $p$ has the property that it is equidistant to the set $S$ (although there may be other points with this property in some metric spaces, or even in $\mathbb{R}^n$ if $S$ has cardinality $n$ or less).