A space is compact if every open cover has a finite subcover. Heine-Borel: in ℝⁿ, compact ⇔ closed and bounded. Extreme-value theorem: continuous f on a compact K attains its max/min.
Compactness
Related concepts
- Open set
- Brouwer fixed-point theorem
- Extreme value theorem
- Heine–Borel theorem
- Stone–Weierstrass theorem
- Tychonoff's theorem
- Radon measure
- Baire category theorem
- Krein-Milman (extreme points)
- Compact operators & Fredholm theory
- Free-electron laser (FEL)
- Bioorthogonal chemistry: k₂ > 1 M⁻¹s⁻¹, zero biological cross-reactivity
- Abiogenesis hypotheses