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Generalizations of Tietze's theorem on convex sets are given in restricted settings. For example Theorem : Let S c Rd be a connected polyhedron locally homeomorphic to the closed d ball. If the points of local nonconvexity of S are decomposable into n convex sets, then S is n + 1 polygonally connected.

On the polygonal connectivity of polyhedra and the closure of open connected sets