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