compactness

# 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 3. **Product Spaces** - Tychonoff: product of compact spaces is compact - Finite products preserve compactness directly 4. **Consequences of Compactness** - Continuous image of compact is compact - Continuous real function on compact attains max/min - `sympy_compute.py maximum "f(x)" --var x --domain "[a,b]"` ## Tool Commands ### Z3_Bounded_Closed ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "bounded_and_closed" ``` ### Sympy_Limit ```bash uv run python -m runtime.harness scripts/sympy_compute.py limit "a_n" --var n --at oo ``` ### Sympy_Maximum ```bash uv run python -m runtime.harness scripts/sympy_compute.py maximum "f(x)" --var x --domain "[a,b]" ``` ## Key Techniques *From indexed textbooks:* - [Topology (Munkres, James Raymond) (Z-Library)] CompactSpaces163 164ConnectednessandCompactnessCh. Itisnotasnaturalorintuitiveastheformer;somefamiliaritywithitisneededbefor...