---- > [!definition] Definition. ([[precompact]]) > A subset $A$ of a [[topological space]] $X$ is said to be **precompact** if its [[closure]] $\overline{A}$ is [[compact]]. ^definition > [!equivalence] Equivalence for Hausdorff spaces. > For $X$ [[Hausdorff space|Hausdorff]], $A \subset X$ is precompact iff it is contained in some [[compact]] $K$. ^equivalence > [!proof]- > $\to$. If $A$ is precompact, then $\overline{A} \supset A$ is [[compact]]; put $K=\overline{A}$. > $\leftarrow.$ Suppose $K \supset A$ for some compact $K$. Since [[every compact subspace of a Hausdorff space is closed]], $K$ is closed, and since $\overline{A}$ is the smallest closed set containing $A$, we attain $A \subset \overline{A} \subset K$. Now use [[closed subspaces of compact spaces are compact]]. > [!equivalence] Equivalence for metrizable spaces. > If $X$ is [[metrizable]], then $A \subset X$ is precompact if and only if every [[sequence]] $(a_{n}) \subset A$ has a [[converge|convergent]] [[subsequence]] in $X$. > > If $X$ is *[[complete]]* and [[metrizable]], then $A$ is [[precompact]] iff it's [[totally bounded]][^5] ^equivalence > [!basicnonexample] > A [[Riesz's Lemma|closed ball is never compact in infinite dimensions]], hence the ball (open or closed) in a [[norm|normed space]] $X$ with infinite dimension is never [[precompact]]. ^nonexample > [!equivalence] Equivalence for *finite-dimensional* normed spaces. (Pre-Heine Borel) > If $(X, \|\cdot\|)$ is a finite-dimensional [[norm|normed space]], then [[Heine-Borel theorem|Heine-Borel]] and the definition above combine to give $\text{precompact} \iff \text{bounded}. \, \, (\text{dim }X<\infty)$ ^equivalence [^5]: This is just because [[for metric spaces, compact iff complete and totally bounded]]; so $\overline{A}$ is [[compact]] iff [[complete]] and [[totally bounded]]; [[complete|completeness]] of $\overline{A}$ always comes [[every compact subspace of a Hausdorff space is closed|for free]] from that of $X$, so [[totally bounded]] is the condition left standing: $\overline{A}$ is [[compact]] iff [[totally bounded]] (this is noting the easy fact that $\overline{A}$ TB iff $A$ TB.) > [!proof] > Recall that [[for metrizable spaces, compact iff limit point compact iff sequentially compact]]. > > Let $X$ be [[metrizable]]. Suppose $A \subset X$ is precompact, and let $(a_{n})$ be a [[subsequence]] of $A$. Then $\overline{A}$ is [[compact]], hence [[sequentially compact]]. So $(a_{n}) \subset A \subset \overline{A}$ has a [[subsequence]] converging to some $a \in \overline{A} \subset X$. > > Conversely, suppose every $(a_{n}) \subset A$ has a convergent subsequence in $X$. It's enough to show $\overline{A}$ is [[sequentially compact]], i.e. that every $(\overline{a}_{n}) \subset \overline{A}$ has a convergent subsequence in $\overline{A}$. For each $\overline{a}_{n}$, [[neighborhood-basis characterization of set closure|define]] a sequence $(a_{n})$ in $A$ by taking $a_{n} \in A$ such that $d(\overline{a}_{n}, a_{n})<1/n$. Extract a [[subsequence]] $(a_{n_{k}})$ converging to some $\overline{a} \in \overline{A}$. Now $d(\overline{a}_{n_{k}}, a_{n_{k}}) \to 0$, hence $\overline{a}_{n_{k}} \to \overline{a}$. ---- #### ---- #### 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 > ```