Having moved through the preceding compactness ideas, we now study compact subspaces and countable covers. The focus keyword for this lesson is compact subspaces and countable covers. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::corollary A closed subset of a compact metric space is compact. ::: :::proof Given that $Y$ is closed in compact $X$. To prove that $Y$ is compact. Add $X\setminus Y$ to any open cover of $Y$ to obtain an open cover of $X$. A finite subcover of $X$ gives a finite subcover of $Y$. $\square$ ::: :::theorem[Compact Subsets of Metric Spaces Are Closed] Every compact subset of a metric space is closed. ::: :::definition[Countably Compact Metric Space] A metric space is $countably compact$ if every countable open cover has a finite subcover. ::: :::theorem[Compactness and Countable Compactness] In metric spaces, compactness is equivalent to countable compactness. ::: :::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 Lebesgue Numbers for Compact Covers] 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/lebesgue-numbers-for-compact-covers button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/sequential-forms-of-compactness :::