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0-regular transformations alternating transformation Cantor set closed set closed subsets closed under continuous collection 8 separates completely non-alternating modulo completely non-alternating transformations completely non-separating modulo connected set contain points contained in T~l(y contains a set contains a2 contains Ti(ai continuous trans continuous transformation contradicts Corollary cut point definition given dendrite doctor of philosophy Duke Mathematical Journal factor theorems following theorems Fundamenta Mathematicae G. T. Whyburn given by Vance homeomorphism interior transformation Let the collection limit point locally connected locally non-alternating transformation MODULO A FAMILY n-coherent necessary and sufficient NON-ALTERNATING AND NON-SEPARATING non-Peanian NON-SEPARATING TRANSFORMATIONS MODULO open set Peano continuum point 61 point of U.a points of T~l(y preceding theorems product and factor Proof separates two points separating modulo set contained set in brackets set T~l(x single point single-point sets space strict sense subcontinuum tained Theorem 5.1 Ti(K Tt is non-alternating TTl(Bt types of transformations unicoherent unless a subset upper semi-continuous