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From Baby Rudin: Let $E$ be a non-empty subset of metric space $X$ and let $S = \{ d(x,y) | x,y \in E \} $ Then the diamater of $E$ is the least upper bound of $S$.

Question: Don't we need that the set $S$ be bounded above in order for its supremum to exist? Or is this implicitly assumed?

Rudin specified that $E$ need be non-empty because or else $S$ would of course be empty. Don't we need to also assume that $S$ is bounded above in order to define the supremum of $S$, which would be the diameter of $E$?

Jeff8770
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  • Short answer: yes. If the set of distances is unbounded you can decide to say the diameter does not exist, or that it's infinite. Either convention will do for any use you want to make of the concept. – Ethan Bolker Nov 11 '23 at 21:55
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    The definition given in Rudin is a little ambiguous. It states that the diameter of $E$ is defined as the least upper bound of $S$, but the concept of upper bounds is only defined relative to the ordered set $Y$ in which $S$ is embedded. Here, it makes sense to let $Y=[0,\infty]$ with the usual ordering. Then, every subset of $Y$ has a least upper bound, and the issue disappears. (We also don't need to make a special allowance for the empty set: if $E=\varnothing$, then $S=\varnothing$, so $\sup S=0$; that is, the diameter of the empty set is $0$.) – Joe Nov 11 '23 at 22:18

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