For any continuous map from a closed triangulable n-manifold to R^n there exists a connected component of f-neighbor pairs that includes both a pair of topologically distant points and a pair of identical points.
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Borsuk-Ulam Type Theorems and Mountain Climbing Problem
For any continuous map from a closed triangulable n-manifold to R^n there exists a connected component of f-neighbor pairs that includes both a pair of topologically distant points and a pair of identical points.