distance between point and empty set

Question

While playing with my little sister earlier we where inventing distances form earth to sun/stars/planets and who had the bigger distance wins. Now at some point she said "from earth to jesus" and since i'm not sure about his existence i was wondering , what if jesus does not exist? Then this distance is the distance form a point on earth to the empty set. Now i was wondering if such a distance is geometrically defined. Is there for example a value or definition for the distance on the real line between a number and the empty set? In euclidian space? In general? I have no knowledge of topology..maybe i could find an answer there?

I hope this is not a silly question , but i want to know who wins (:

Answer 1

In general, only the distance between two points (in euclidean space or more generally in a metric space) is defined.

This notion can be extended to define the distance between two subsets $A.B$ of a metric space as $$d(A,B):=\inf\{\,d(a,b)\mid a\in A, b\in B\,\}.$$ But take care! It may happen that $d(A,B)=0$ even when $A\ne B$ (for example when $A\cap B$ is not empty), a property that would not be desired for a metric (read: it is not fully justified to call this $d$ a distance). Anyway, this extended notion of distance between two sets would take the infimum of the empty set if $A$ or $B$ are empty. By careful definition, $\inf\emptyset$ should be the greatest real number that is a lower bound of $\emptyset$. However, every real number is a lower bound for $\emptyset$ and there is no greatest real number. Hence in a strict sense $\inf\emptyset$ is not defined (you win!). However, usually one extends the definition of infimum to allow two special values, namely $-\infty$ for a set that has no lower bound and $+\infty$ for the empty set. In this set, $d(A,\emptyset)=\inf\emptyset=+\infty$ (and you lose!)

So in a way there's atie between you two players (and isn't that WJWD?) On the other hand, as a Christian believer, your sister could not claim the win as God (and hence Jesus) is omnipresent, hence at distance $0$ from anywhere. :)

Answer 2

The typical definition of a distance function from a point $p$ to a set $S$ looks roughly like this:

$$d(p, S) = \inf\{d(p,s): s\in S\}$$

If $S$ is the empty set, then $d(p, S)=\inf\varnothing$. Under the usual rules of how to define a distance function, this is undefined.

However, there is a strong case that it "should be" $\infty$. It should be clear (after some thinking) that if $A\subseteq B$ then $d(p, B)\leq d(p, A)$ because $B$ simply has more points to choose from; that could decrease the minimum but it cannot possibly increase the minimum. Since $\varnothing$ is a subset of every set, it must be that $d(p,A)$ is greater than [or equal to] $d(p,B)$ for any set in the space. If the space is unbounded, then $d(p,B)$ can be arbitrarily large; so we need a value greater than any number— this is the definition of $\infty$.

[If the space is bounded, then we could do with smaller values, but we would never need a larger value.]

In short, you lose :)

Technical details: this all occurs in metric spaces which are a type of topological space where distance is defined in an intuitive way. In particular, all manifolds are metric spaces, so basically any physical model of the spacial dimensions of the universe. There are more general notions of distance that I am unfamiliar with, perhaps this intuition breaks down there.