Let $(M,d)$ be a [metric space](/page/Metric%20Space). Let $\mathbb{N}:=\{1,2,3,\dots\}$. For $a \in M$ and $r>0$, write $B(a,r):=\{z \in M : d(z,a)<r\}$ for the open ball in the metric topology induced by $d$. Suppose compactness means every [open cover](/page/Open%20Cover) in this topology has a [finite subcover](/page/Finite%20Subcover), completeness means every $d$-[Cauchy sequence](/page/Cauchy%20Sequence) in $M$ converges to a point of $M$, and [total boundedness](/page/Total%20Boundedness) means that for every $\varepsilon > 0$ there exists a finite subset $F \subset M$ such that $M \subset \bigcup_{a \in F} B(a,\varepsilon)$. Then $(M,d)$ is compact if and only if it is complete and [totally bounded](/page/Totally%20Bounded).