Having moved through the preceding compactness ideas, we now study Lebesgue numbers for compact covers. The focus keyword for this lesson is Lebesgue numbers for compact covers. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::definition[Lebesgue Number] Let $\{G_\lambda:\lambda\in\Lambda\}$ be an open cover of $X$. A number $\delta>0$ is a $Lebesgue number$ if for every $x\in X$, some $G_\lambda$ satisfies $S(x,\delta)\subseteq G_\lambda$. ::: :::theorem[Lebesgue Number Lemma] Every open cover of a compact metric space has a Lebesgue number. ::: :::proof Given that $X$ is compact and $\{G_\lambda\}$ is an open cover. To prove that a Lebesgue number exists. Around each $x\in X$, choose a ball $S(x,2r_x)$ contained in one member of the cover. Finitely many balls $S(x_i,r_{x_i})$ cover $X$. The minimum of the finitely many radii is a positive Lebesgue number. $\square$ ::: :::example[Uniform Cover Control] A Lebesgue number gives one radius that works everywhere, although the cover member may depend on the centre of the ball. ::: :::exercise [1] State the main definition or theorem from this lesson. [2] Give one example where the hypothesis is used. [3] Explain one common mistake related to this topic. ::: :::answer [1] See the first formal block of the lesson. [2] The examples in the lesson provide standard choices. [3] A common mistake is to apply a Euclidean compactness rule in an arbitrary metric space. ::: :::faq[Frequently Asked Questions] Q: Why is the metric important? A: Compactness, boundedness, convergence, and openness are all interpreted using the chosen metric. Q: What should students check first? A: Identify the space, the metric, and the exact compactness criterion being used. ::: :::call-to-action[Continue to Compactness Under Continuous Mappings] subtitle: Review the definitions, proofs, and examples before moving to the next compactness idea. button: Next Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/compactness-under-continuous-mappings button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/compact-subspaces-and-countable-covers :::