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Allen Hatcher Homepage Prompt AI with the Theorem and Related Material: Example Theorem (plus discussion and proof, not shown): Theorem 1.10. For every continuous map f : S 2→R 2 there exists a pair of antipodal points x and −x in S 2 with f (x) = f (−x). ( Algebraic Topology,  33-34) Prompt: Illustrate the following proof:  Theorem 1.10. For every continuous map f : S 2→R 2 there exists a pair of antipodal points x and −x in S 2 with f (x) = f (−x).  AI The provided text is Allen Hatcher's classic algebraic topology proof of the Borsuk–Ulam Theorem for \(n = 2\). Below is an breakdown of the proof's core mechanics alongside a visual simulation of the path lifting argument that drives the contradiction. [ 1 , 2 ] 1. Conceptual Framework The proof operates by contradiction via three primary geometric mappings: Equator Loop Odd Map Covering Map I -------------> S² -------------------------> S¹ <------------------------ ℝ ...