Having moved through the preceding compactness ideas, we now study compactness under continuous mappings. The focus keyword for this lesson is compactness under continuous mappings. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::theorem[Continuous Image of a Compact Space] Let $X$ be compact and let $f:X\to Y$ be continuous. Then $f(X)$ is compact. ::: :::proof Given that $X$ is compact and $f$ is continuous. To prove that $f(X)$ is compact. Let $\mathcal{G}$ cover $f(X)$. The inverse images $f^{-1}(G)$ cover $X$ and are open. A finite subcover of $X$ maps forward to a finite subcover of $f(X)$. $\square$ ::: :::corollary Compactness is preserved under homeomorphism. ::: :::solved-problem If $f:X\to X$ is continuous on compact $X$ and $f(X)$ is dense in $X$, then $f(X)=X$. ::: :::solution The set $f(X)$ is compact and hence closed. Since it is also dense, it equals its closure, which is $X$. $\square$ ::: :::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 Extreme Values on Compact Domains] 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/extreme-values-on-compact-domains button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/lebesgue-numbers-for-compact-covers :::