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 $\varnothing$?
Best Answer
The following books explicitly take the position that $\operatorname{diam}\varnothing =0$:
The following books explicitly take the position that $\operatorname{diam}\varnothing =-\infty$:
(I never heard of either of these before Google Books search brought them up.)
The following books explicitly restrict the definition of diameter to nonempty sets:
It seems that W. Sierpiński, General topology, belongs to the second or third category, because the author says on page 110: "Thus the diameter of every non-empty set contained in a metric space is a uniquely defined real non-negative number, finite or infinite". But it's not very clear what Sierpiński's intention was when writing this.
Many books do nothing of the above: they define the diameter of a set as supremum of pairwise distances, and offer no further details.
The definition of Hausdorff measure would become awkward. For example, the 1-dimensional measure involves the infimum of $\sum \operatorname{diam} U_i$ over certain families of sets. If $\operatorname{diam}\varnothing =-\infty$, we'd be able to make the infimum $-\infty$ by throwing in the empty set. (Note that the Wikipedia article explicitly says that $\operatorname{diam}\varnothing =0$). One can try to fix this by requiring $U_i$ to be nonempty, but then the measure of empty space becomes a special case (and the measure of $\varnothing$ definitely needs to be $0$).
Another issue is the inequality $$ \operatorname{diam}(A\cup B)\le \operatorname{diam}A+\operatorname{diam}B+\operatorname{dist}(A,B) $$ which should hold for all $A,B$. Suppose $B$ is empty but $A$ is not. The right-hand side becomes undefined due to presence of $\operatorname{diam}\varnothing =-\infty$ and $\operatorname{dist}(A,\varnothing)=+\infty$. (And the latter definitely needs to be $+\infty$.)
Third issue: if one applies a metric transform, i.e., replaces metric $d$ with $\varphi(d)$ where $\varphi $ is an increasing concave function, the diameters of sets should transform accordingly. With $-\infty$ in the mix, one is led to awkward conventions ($\sqrt{-\infty}=-\infty$?).
That said, I can imagine some arguments in favor of $\operatorname{diam}\varnothing =-\infty$. One is that the following statement becomes true:
(quoted from S. Willard, General topology). If $\operatorname{diam}\varnothing =0$, the above is false without additional requirement that $C_n$ are nonempty.
That said, it's probably best to put nonempty there. The absence of nonempty leads to wrong statements in a number of books, e.g., "if $N$ is compact, there exist $x,y\in N$ such that $\rho(x,y)=\operatorname{diam}N$". (G.T. Whyburn, Analytic topology).
Summary.