---- > [!definition] Definition. ([[totally bounded]]) > A [[metric space]] $(X, d)$ is said to be **totally bounded** if for every $\varepsilon>0$ there is a finite [[cover|covering]] of $X$ by $\varepsilon$-[[metric topology|balls]]. > ^definition > [!note] Remarks. > - Clearly [[compact]] $\implies$ totally bounded. The [[for metric spaces, compact iff complete and totally bounded|converse]] holds if $X$ is [[complete]]. > - Clearly totally bounded $\implies$ [[bounded set|bounded]]. The converse holds e.g. for subsets of finite-dimensional [[norm|normed spaces]].[^1] ^note [^1]: Indeed, [[norm|all norms on finite-dimensional vector spaces are topologically equivalent]] so WLOG consider $(\mathbb{R}^{n}, \|\cdot\|_{\infty})$ and a bounded subset $S$. Using boundedness, fix a cube $Q=[-R,R]^{n} \supset S$. Cleary $Q$ can covered by finitely many $\varepsilon$-cubes, thus so can be $S$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```