Boundedness in Metric Spaces#

Theorem: Boundedness and Distance

Let \((M, d)\) be a metric space with its metric topology \(\tau_d\).

A subset \(S \subseteq M\) is bounded if and only if there exists some \(r \in \mathbb{R}_{\ge 0}\) such that \(d(x,y) \lt r\) for all \(x,y \in S\).