TITLE: What is the best way to define the diameter of the empty subset of a metric space?
QUESTION [16 upvotes]: This question is related to Why are metric spaces non-empty? .  I think that a metric space should allowed to be empty, and many authorities, including Rudin, agree with me.  That way, any subset of a metric space is a metric space, you don't have to make an exception for $\varnothing$, and you can ascribe certain properties to $\varnothing$, such as, $\varnothing$ is compact and connected. By definition of diameter of a metric space, the diameter of $\varnothing$ should be $-\infty$, since the $\sup$ of the empty set is $-\infty$.  This "feels wrong", since the diameter is a measure of the "size" of a metric space or a subset thereof, and its seems like the diameter of $\varnothing$ ought to be zero.  
I guess I really have two questions:
If you allow the diameter of the empty set to be $-\infty$, does it lead to problems?  For example, $-\infty$ plus any real number is $-\infty$, and I could imagine how that might lead to a problem, but I haven't seen an actual situtation where that happens.
In practice, what do expert analysts (such as Rudin, Folland, Royden, etc.) use for the diameter of $\varnothing$?

REPLY [4 votes]: I've been reading through Kolmogorov & Fomin, Introductory Real Analysis, which has the following two problems at the end of section 7 (note: $\rho(\cdot,\cdot)$ is the distance in the metric space):

Problem 4. By the diameter of a subset $A$ of a metric space $R$ is meant the number
  $$
d(A) = \sup_{x,y\in{A}} \rho(x,y).
$$
  Suppose R is complete, and let ${A_n}$ be a sequence of closed subsets of $R$ netsted in the sense that 
  $$
A_1 \supset A_2 \supset \ldots \supset A_n.
$$
  Suppose further that
  $$
\lim_{n\rightarrow{}\infty}d(A_n) = 0.
$$
  Prove that the intersection $\bigcap\limits_{n=1}^{\infty}A_n$ is nonempty.



Problem 6. Give an example of a complete metric space $R$ and a nested sequence ${A_n}$ of closed subsets of $R$ such that
  $$
\bigcap\limits_{n=1}^{\infty}A_n = \emptyset.
$$
  Reconcile this example with Problem 4.

To answer Problem 6:
If $R=\emptyset$, then $R$ is a complete metric space. And if we define $A_n=\{\emptyset,\emptyset,\ldots\}$, then $A_n$ is a nested sequence of closed subsets of $R$. And clearly $\cap_{n=1}^{\infty}A_n = \emptyset$. In order to reconcile this with Problem 4 we may well ask, what is the diameter of the empty set? As suggested by @user111213, this cannot be reconciled if the empty set has diameter zero. If the diameter is $-\infty$, or if it is not defined, then the limit in problem 4 cannot be zero, thus reconciling our example.