TITLE: Does Rudin's definition of "bounded" overlook the empty set?
QUESTION [6 upvotes]: In Definition $2.18$ of Baby Rudin, Rudin defines boundedness for metric spaces as follows: given a metric space $X$, and a set $E \subset X$, we say $E$ is bounded if there exists a real number $M$ and a point $q \in X$ such that $d(p,q) < M$ for all $p \in E$. 
Per this definition, if $E=X=\varnothing$, then $E$ is unbounded. The reason is that "there exists a point $q \in X$" is a false statement if $X = \varnothing$. 

Should this be regarded as a minor mistake/typo in Baby Rudin? Or are there particular reasons the empty set should be considered unbounded?

If we use the alternative definition that a set $E$ is bounded if there exists a real number $M$ such that for all $p,q \in E$, $d(p,q) < M$, then this coincides with the above definition for nonempty sets, but also always considers the empty set bounded.

REPLY [9 votes]: This definition is not standard and should be considered as an error in Baby Rudin.  There is no reason whatsoever to consider the empty set unbounded in the empty space, and Rudin almost certainly did not intend to do so.  Indeed, I would not be surprised if there are some theorems later in the text that are incorrect with his definition.