Having moved through the preceding compactness ideas, we now study epsilon nets and total boundedness. The focus keyword for this lesson is epsilon nets and total boundedness. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::definition[Epsilon-Net] Let $(X,d)$ be a metric space and let $\varepsilon>0$. A subset $A\subseteq X$ is an $epsilon-net$ for $X$ if $$\begin{equation*} X=\bigcup_{a\in A}S(a,\varepsilon). \end{equation*}$$ ::: :::definition[Totally Bounded Metric Space] A metric space $X$ is $totally bounded$ if for every $\varepsilon>0$ there exists a finite epsilon-net for $X$. ::: :::lemma Every subset of a totally bounded metric space is totally bounded. ::: :::proof Given that $X$ is totally bounded and $Y\subseteq X$. To prove that $Y$ is totally bounded. Cover $X$ by finitely many balls of radius $\varepsilon/2$. From every such ball that meets $Y$, choose one point of $Y$. These finitely many chosen points form an epsilon-net for $Y$. $\square$ ::: :::theorem[Total Boundedness Implies Boundedness] Every totally bounded metric space is bounded. ::: :::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 Total Boundedness and Cauchy Subsequences] 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/total-boundedness-and-cauchy-subsequences button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/closedness-and-boundedness-of-compact-sets :::