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Definition : A set $S$ is said to be bounded in totally ordered set $X$ if there exist $M \in X$ such that $a \lt M $ for all $ a$ $\in S $ .

Definition : A set $S$ is said to be bounded in a metric space $(X,d)$ if $diam(S) < \infty$. Where $diam(S) = sup\{d(a,b):a,b \in S\}$

What is difference between these two definitions, and if I'm dealing with convergence in a space than which definition should be used ?

For example, if I say a sequence is bounded than what does it mean in arbitrary metric spaces ?

Tutankhamun
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1 Answers1

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the diameter map $$diam_d:\mathfrak{P}(X) \to \{x \in \mathbb{R}|x \ge 0 \} \cup \{\infty\}$$ maps the power set of the underlying set of the metric space $(X,d)$ to the totally ordered set consisting of the non-negative reals augmented by plus infinity - which we may call $[0,\infty]$

$S \subseteq X$ is bounded in $(X,d)$ iff $diam(S) \lt \infty$

looking to the definition of diameter, if we look at the set $$S' \subseteq [0,\infty] = \{d(x,y)|x,y \in S\}$$ we see that $S$ is bounded in $(X,d)$ iff $S'$ is bounded in $[0,\infty]$

David Holden
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