open-sets

# Open Sets ## When to Use Use this skill when working on open-sets problems in topology. ## Decision Tree 1. **Is f: X -> Y continuous?** - For metric spaces: x_n -> x implies f(x_n) -> f(x)? - For general spaces: f^(-1)(open) = open? - For products: Check each coordinate function - `z3_solve.py prove "preimage_open"` 2. **Open Set Verification** - For metric spaces: for all x in U, exists epsilon > 0 with B(x,epsilon) subset U - `z3_solve.py prove "ball_contained"` with epsilon witnesses 3. **Topological Properties** - Interior: int(A) = largest open subset of A - Closure: cl(A) = smallest closed superset of A - Boundary: bd(A) = cl(A) \ int(A) 4. **Continuity Tests** - Epsilon-delta: for all epsilon > 0, exists delta > 0: d(x,a) < delta implies d(f(x),f(a)) < epsilon - `z3_solve.py prove "epsilon_delta_bound"` ## Tool Commands ### Z3_Preimage_Open ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "preimage_open" ``` ### Z3_Epsilon_Delta ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "ForAll(eps, Exists(delta, d(x,a) < delta implies d(f(x),f(a)) < eps))" ``` ### Z3_Ball_Contained ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "ball_contained" ``` ## Key Techniques *From indexed textbooks:* - [Introduction to Topological Manifolds... (Z-Library)] Show that every local homeomorphism is an open map. Show that every homeomor...