Definition:Bounded Metric Space/Euclidean

Definition

Let $A \subseteq \R^n$ be a subset of a Euclidean space under the usual metric.

$A$ is bounded (in $\R^n$) if and only if :

$\exists N \in \R: \forall x \in A: \size x \le N$

That is, every element of $A$ is within a finite distance of any point we may choose for the origin.

Unbounded

Let $A \subseteq \R^n$ be a subset of a Euclidean space under the usual metric.

$A$ is unbounded (in $\R^n$) if and only if $A$ is not bounded (in $\R^n$).

Also see

• Results about bounded Euclidean spaces can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bounded set
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bounded set