# Distance from a point to empty set.

Let $(X,d)$ be a metric space and let $A \subseteq X$. We define the distance from a point $x \in X$ to $A$ by $d(x,A)= \inf \{ d(x,a) : a \in A \}$.

What will be the value of $d(x, \emptyset )$? I am confused between $+ \infty$ and $- \infty$. Also, is it possible to find $\min \{ d(x,a) : a \in \emptyset \}$? (I know $\emptyset$ is empty, just asking symbolically whether we can take $\min$ instead of $\inf$)

Thanks.

• I think the set $\{d(x,a):a\in A\}$ is itself an empty set. So in that context this question might help. But I could be wrong. – R_D Apr 23 '16 at 6:33
• Clearly $-\infty$ is impossible since $d:X^2\to [0,\infty)$... – BigbearZzz Apr 23 '16 at 6:34
• @Rise I don't understand, why will it be an empty set? – deditus Apr 23 '16 at 6:37
• @BigbearZzz I think $d$ is not necessarily bounded below by $0$ because diameter of an empty set is $- \infty$. In diameter, we take $sup$ instead of $inf$ – deditus Apr 23 '16 at 6:44
• @deditus, since you are considering the set of distances from x to no point there are no distances to consider and hence the set is empty. That was my (possibly flawed) logic. – R_D Apr 23 '16 at 6:44

$d(x,\emptyset)=\inf\emptyset=\infty.$ That makes sense because in general, if $A\subseteq B,$ then $d(x,A)\ge d(x,B);$ so $d(x,\emptyset)\ge d(x,B)$ for all $B.$

If the infimum of a set $S$ is attained, that is, if $\inf S\in S,$ then $\min S=\inf S;$ otherwise, $\min S$ does not exist. Thus $\min\emptyset$ does not exist, just as $\min\{\frac1n:n\in\mathbb N\}$ does not exist. It should be no surprise that the empty set has no least element, seeing as it has no elements at all.

Fix $x \in X$. If $\{d(x,a) \mid a \in \emptyset\} \neq \emptyset$, then there exists $a \in \emptyset$.

It is vacuously true that for each $M \in \mathbb{R}$ we have $a \geq M$, for all $a \in \emptyset$, so it's reasonable to define $inf\{\emptyset\}=\infty$.

The $min$ function returns a minimal element contained in a set for which an order relation is defined, if such an element exists, so it's again reasonable to say $min\{\emptyset\}=\emptyset$.

If $A$ is empty, then $\{ d(x,a) : a \in A \}$ is empty. Then inf$\{\emptyset \} = \infty$.

The proof that inf$\{\emptyset \} = \infty$ can be found in this post:

https://math.stackexchange.com/a/432308