----- > [!proposition] Proposition. ([[Lebesgue number lemma]]) > Let $\mathscr{A}$ be an [[cover|open covering]] of the [[metric space]] $(X,d)$. If $X$ is [[compact]], then there exists $\delta>0$ s.t. for each subset of $X$ having [[diameter of a bounded subset of a metric space|diameter]] less than $\delta$, there is an element of $\mathscr{A}$ containing it. > > > [!definition] Definition. (Lebesgue number) > > $\delta$ is called the **Lebesgue number** for the covering $\mathscr{A}$. ^definition > [!proof]- Proof. ([[Lebesgue number lemma]]) > Using that $X$ is [[compact]], obtain a finite subcover $\{ A_{1},\dots ,A_{n} \}$ of $X$. For each $i$, set $C_{i}:=X-A_{i}$, then define $f:X \to \mathbb{R}$ by averaging over the [[distance from point to set|distances]] $d(x,C_{i})$ thus: $f(x)=\frac{1}{n} \sum_{i=1}^{n}d(x,C_{i}).$ Note $f$ is [[continuous]] as a some thereof. It is strictly positive (i.e., no $x \in X$ can live simultaneously in all $C_{i}$) also: given $x \in X$ and choosing $i$ so that $x \in A_{i}$, there exists $\varepsilon>0$ such that $x \in B_{\varepsilon}(x) \subset A_{i}.$ So $d(x,C_{i}) \geq \varepsilon$ and therefore $f(x) \geq \frac{\varepsilon}{n}$. > By the [[extreme value theorem]], $f$ attains a [[global extrema|minimum value]] $\delta$ on $X$; we claim $\delta$ is our desired Lebesgue number. Let $B \subset X$ have [[diameter of a bounded subset of a metric space|diameter]] less than $\delta$. Let $b \in B$, then $b \in B_{\delta}(b)$. Now $\delta \leq f(b) \leq d(b, C_{m}),$ where $C_{m}$ is the largest of the numbers $d(x_{0},C_{i})$. Then $B_{\delta}(b)$ is contained in the element $A_{m}=X-C_{m}$ of the covering $\mathscr{A}$. ----- #### ---- #### 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 > ``` #reformatrevisebatch04