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Intuitively, a [[set]] is '''bounded''' if the [[distance]]s between its points are all less than some [[finite]] [[real number]] (the ''bound'').  Formally, we say that a [[subset]] <math>S</math> of a [[metric space]] <math>(X, d)</math> (such as the standard [[Euclidean plane]], <math>\mathbb{R}^2</math> with distance <math>d((x, y), (w, z)) = \sqrt{(x - w)^2 + (y - z)^2}</math>), is bounded if for some <math>x \in X</math> there exists some <math>M \in \mathbb{R}_{\geq 0}</math> such that for all <math>s \in S</math>, <math>d(s, x) < M</math>.

Note that if a set <math>S</math> is bounded, the choice of <math>x</math> is immaterial if are willing to change the bound: we have by the [[triangle inequality]] that <math>d(y, s) \leq d(y, x) + d(x, s) \leq d(x, y) + M</math> for all <math>s \in S</math>.

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[[Category:Topology]]