Compactness implies total boundedness

In a metric space, every compact set can be covered by finitely many small balls.

Compactness implies total boundedness

Compactness implies total boundedness: Let $(X,d)$ be a metric space and let $K\subseteq X$ be compact . Then $K$ is totally bounded : for every $\varepsilon>0$ there exists a finite set $F\subseteq K$ such that

K\subseteq \bigcup_{x\in F} B(x,\varepsilon),

where $B(x,\varepsilon)$ denotes the open ball .

This is one half of the standard metric characterization compact iff complete and totally bounded , complementing compactness implies completeness and closely related to the existence of finite epsilon-nets .