Can the distance between 2 non-empty sets be infinite?

Intuitively I would immediately assume no, but that's not how things usually work in math and considering there are different kinds of infinities I haven't been able to find the answer.

Here's my definition of the distance between 2 sets:

$d(A,B) = \inf{\{||\vec a - \vec b||:\vec a \in A, \vec b \in B\}}$

• What is a distance for two general sets? Or, do you mean sets in $\Bbb R^n$?
– Aahz
Aug 20, 2014 at 10:33
• This was a question on my exam. The title was all the given information and we simply had to answer yes or no. Though 'our' definition of distance was: $d(A,B) = \inf{\{||\vec a - \vec b||:\vec a \in A, \vec b \in B\}}$ Aug 20, 2014 at 10:42
• It depends on whether “your” definition of a metric space admits infinite distances. Aug 20, 2014 at 12:42
• Please add the definition of "distance between sets" to the question. I'd do it myself, but you've got the lovely TeX all ready to go in your comment. Aug 20, 2014 at 15:38

$$d(A,B) = \inf_{x\in A,\ y\in B} d(x,y)$$
That means that if $x\in A$ and $y\in B$ then $d(x,y) \geq d(A,B)$.
Now, $d(x,y)$ is always finite in a metric space so $d(A,B)$ must be too.
Let $x$ be an element of $A$ and let $y$ be an element of $B$. We know that $\|x-y\|$ is a real number $r$ and so $d(A,B)$ must be at most $r$ by the definition of $d(A,B)$, hence $d(A,B)$ is finite.