----- > [!proposition] Proposition. ([[for metric spaces, compact iff complete and totally bounded]]) > Let $(X,d)$ be a [[metric]] [[topological space|space]]. We know [[for metrizable spaces, compact iff limit point compact iff sequentially compact|compact iff]] [[sequentially compact|sequentially compact for]] [[metric space|metric spaces]] and [[Cauchy sequence is convergent iff has convergent subsequence]], hence any [[compact]] [[subspace topology|subspace]] $K \subset X$ is [[complete]]. > A necessary and sufficient condition for the converse to hold is to require $K$ to moreover be [[totally bounded]]. That is: > $K \subset X \text{ is compact } \iff K \text{ is complete and totally bounded}.$ ^proposition > [!specialization] > Since [[complete|closed iff complete]] and [[totally bounded|totally bounded iff bounded]] for [[subspace topology|subspaces]] of $\mathbb{R}^{n}$, one recovers [[Heine-Borel theorem|Heine-Borel]] from this result. ^specialization > [!proof]- Proof. ([[for metric spaces, compact iff complete and totally bounded]]) > ~ ----- #### ---- #### 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 > ```