connectedness

# Connectedness ## When to Use Use this skill when working on connectedness problems in topology. ## Decision Tree 1. **Is X connected?** - Strategy 1 - Contradiction: * Assume X = U union V where U, V are disjoint, non-empty, and open * Derive a contradiction - Strategy 2 - Path connectedness: * Show for all x,y in X, exists continuous path f: [0,1] -> X with f(0)=x, f(1)=y - Strategy 3 - Fan lemma: * If {A_i} are connected sharing a common point, then union A_i is connected 2. **Connectedness Proofs** - Show no separation exists - `z3_solve.py prove "no_separation"` - Use intermediate value theorem for R subsets 3. **Path Connectedness** - Construct explicit path: f(t) = (1-t)x + ty for convex sets - `sympy_compute.py simplify "(1-t)*x + t*y"` to verify path 4. **Components** - Connected component: maximal connected subset containing x - Path component: maximal path-connected subset containing x ## Tool Commands ### Z3_No_Separation ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "no_separation" ``` ### Sympy_Path ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "(1-t)*x + t*y" ``` ### Z3_Ivt ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "intermediate_value" ``` ## Key Techniques *From indexed textbooks:* - [Introduction to Topological Manifolds... (Z-Library)] Connectedness One of the most important elementary facts about continuous functions ...