What is the best way to define the diameter of the empty subset of a metric space?
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 $\varnothing$?
The following books explicitly take the position that $\operatorname{diam}\varnothing =0$:
- C. Kuratowski, Topology, vol.I
- M. H. A. Newman, Elements of the topology of plane sets of points
The following books explicitly take the position that $\operatorname{diam}\varnothing =-\infty$:
- G. F. Simmons, Introduction to topology and modern analysis
- M. Ó. Searcóid, Metric spaces
(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:
- W. Rudin, Principles of mathematical analysis
- H. L. Royden, Real analysis.
- K. Falconer, Fractal geometry
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.
If you allow the diameter of the empty set to be $−\infty$, does it lead to problems?
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:
In a complete metric space, each decreasing sequence of closed sets $C_n$ with $\operatorname{diam}C_n\to 0$ has nonempty intersection.
(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.
- It's safer to keep $\operatorname{diam}$ nonnegative, because it may appear in formulas that need nonnegative inputs.
- If the validity of what you write depends on the interpretation of $\operatorname{diam}\varnothing$, consider changing the statement.
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.