Totally bounded set

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In mathematics, a totally bounded set is any subset of a metric space with the property that for any positive radius r>0 it is contained in some union of a finite number of "open balls" of radius r. In a finite dimensional normed space, such as the Euclidean spaces, total boundedness is equivalent to boundedness.

Formal definition

Let X be a metric space. A set $A \subset X$ is totally bounded if for any real number r>0 there exists a finite number n(r) (that depends on the value of r) of open balls of radius r, $B_r(x_1),\ldots,B_r(x_{n(r)})\,$ , with $x_1,\ldots,x_{n(r)} \in X$ , such that $A \subseteq \cup_{k=1}^{n(r)}B_r(x_{k})$ .