# Totally bounded set

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Let X be a metric space. A set ${\displaystyle A\subset X}$ 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, ${\displaystyle B_{r}(x_{1}),\ldots ,B_{r}(x_{n(r)})\,}$, with ${\displaystyle x_{1},\ldots ,x_{n(r)}\in X}$, such that ${\displaystyle A\subseteq \cup _{k=1}^{n(r)}B_{r}(x_{k})}$.