Compactness When to Use Use this skill when working on compactness problems in topology. Decision Tree 1. Is X compact? - If X subset R^n: Is X closed AND bounded? (Heine-Borel) - If X is metric: Does every sequence have convergent subsequence? - General: Does every open cover have finite subcover? - 2. Compactness Tests - Heine-Borel (R^n): closed + bounded = compact - Sequential: every sequence has convergent subsequence - to check convergence 3. Product Spaces - Tychonoff: product of compact spaces is compact - Finite products preserve compactness directly 4. Consequences of Compactness -…