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?
   • z3_solve.py prove "bounded_and_closed"

2. Compactness Tests

   • Heine-Borel (R^n): closed + bounded = compact
   • Sequential: every sequence has convergent subsequence
   • sympy_compute.py limit "a_n" --var n to check convergence