Having moved through the preceding compactness ideas, we now study open covers and compactness. The focus keyword for this lesson is open covers and compactness. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::definition[Open Cover] Let $(X,d)$ be a metric space and let $Y\subseteq X$. A collection $\mathcal{G}$ of open subsets of $X$ is called an $open cover$ of $Y$ if $$\begin{equation*} Y\subseteq \bigcup_{G\in\mathcal{G}}G. \end{equation*}$$ Equivalently, every point of $Y$ belongs to at least one member of the collection. ::: :::definition[Finite Subcover] Let $\mathcal{G}$ be an open cover of $Y$. A finite subcollection $\{G_1,G_2,\ldots,G_n\}$ of $\mathcal{G}$ is called a $finite subcover$ of $Y$ if $$\begin{equation*} Y\subseteq \bigcup_{i=1}^{n}G_i. \end{equation*}$$ ::: :::definition[Compact Metric Space] Let $(X,d)$ be a metric space. Then $X$ is called $compact$ if every open cover of $X$ has a finite subcover. ::: :::example[Open Interval Is Not Compact] For $n\geq2$, put $G_n=(1/n,1)$. Then $\{G_n:n\geq2\}$ covers $(0,1)$, but every finite subcollection misses points close to $0$. Hence $(0,1)$ is not compact. ::: :::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 Finite Intersection Tests for Compactness] 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/finite-intersection-tests-for-compactness button-ghost: Previous Lesson | /functional-analysis/metric-spaces/topological-structure-of-metric-spaces/baire-category-theorem-and-applications :::