Having moved through the preceding compactness ideas, we now study finite intersection tests for compactness. The focus keyword for this lesson is finite intersection tests for 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[Finite Intersection Property] Let $X$ be a set and let $\mathcal{F}$ be a collection of subsets of $X$. The collection has the $finite intersection property$ if every finite subcollection has nonempty intersection: $$\begin{equation*} F_1\cap F_2\cap\cdots\cap F_n\neq\varnothing. \end{equation*}$$ ::: :::theorem[Closed-Set Intersection Test] Let $(X,d)$ be a metric space. Then $X$ is compact if and only if every collection of closed subsets of $X$ having empty total intersection has a finite subcollection having empty intersection. ::: :::proof Given that $(X,d)$ is a metric space. To prove that compactness is equivalent to the closed-set intersection condition. Assume $X$ is compact and $\bigcap_{\lambda\in\Lambda}F_\lambda=\varnothing$ for closed sets $F_\lambda$. Then $\{X\setminus F_\lambda:\lambda\in\Lambda\}$ is an open cover of $X$, so finitely many of them cover $X$. Taking complements gives a finite empty intersection. Conversely, apply the same complement argument to an arbitrary open cover. $\square$ ::: :::solved-problem Show that $\mathbb{R}$ is not compact by using the finite intersection property. ::: :::solution Let $F_n=[n,\infty)$. Every finite intersection is nonempty, but $$\begin{equation*} \bigcap_{n=1}^{\infty}F_n=\varnothing. \end{equation*}$$ Therefore $\mathbb{R}$ is not compact. $\square$ ::: :::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 Closedness and Boundedness of Compact Sets] 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/closedness-and-boundedness-of-compact-sets button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/open-covers-and-compactness :::