Having moved through the preceding compactness ideas, we now study closedness and boundedness of compact sets. The focus keyword for this lesson is closedness and boundedness of compact sets. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::definition[Bounded Metric Space] Let $(X,d)$ be a metric space. Then $X$ is $bounded$ if there exists $M>0$ such that $d(x,y)\leq M$ for all $x,y\in X$. ::: :::definition[Diameter of a Set] For a nonempty subset $E$ of a metric space, define $$\begin{equation*} \operatorname{diam}(E)=\sup\{d(x,y):x,y\in E\}. \end{equation*}$$ The set is bounded if its diameter is finite. ::: :::theorem[Compact Subsets Are Closed and Bounded] Let $(X,d)$ be a metric space. If $Y\subseteq X$ is compact, then $Y$ is closed and bounded. ::: :::proof Given that $Y$ is compact. To prove that $Y$ is closed and bounded. Balls $S(y_0,n)$ centred at one fixed point cover $Y$, so finitely many give boundedness. If $x\notin Y$, compactness lets finitely many neighbourhoods of points of $Y$ be chosen so that a small ball around $x$ misses all of $Y$. Hence the complement of $Y$ is open, so $Y$ is closed. $\square$ ::: :::example[Closed and Bounded Need Not Be Compact] An infinite set with the discrete metric is bounded and closed, but the singleton cover has no finite subcover. Thus closed and bounded need not imply compact in arbitrary metric spaces. ::: :::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 Epsilon Nets and Total Boundedness] 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/epsilon-nets-and-total-boundedness button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/finite-intersection-tests-for-compactness :::