Having moved through the preceding compactness ideas, we now study compact domains and homeomorphisms. The focus keyword for this lesson is compact domains and homeomorphisms. 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 Bijection from a Compact Space] Let $X$ be compact and let $f:X\to Y$ be a continuous bijection into a metric space. Then $f^{-1}$ is continuous, so $f$ is a homeomorphism. ::: :::proof Given that $X$ is compact and $f$ is a continuous bijection. To prove that $f^{-1}$ is continuous. If $F$ is closed in $X$, then $F$ is compact. Hence $f(F)$ is compact and therefore closed in $Y$. Since $(f^{-1})^{-1}(F)=f(F)$, inverse images of closed sets under $f^{-1}$ are closed. Thus $f^{-1}$ is continuous. $\square$ ::: :::example[Compactness Is Essential] The map $x\mapsto e^{2\pi ix}$ from $[0,1)$ onto the unit circle is continuous and bijective, but its inverse is not continuous. ::: :::solved-problem The open unit disc and closed unit disc are not homeomorphic. ::: :::solution The closed unit disc is compact and the open unit disc is not. A homeomorphism preserves compactness, so no such homeomorphism exists. $\square$ ::: :::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 Uniform Continuity 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/uniform-continuity-on-compact-domains button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/extreme-values-on-compact-domains :::