## A refinement of the Whyburn cyclic element theory |

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A-analogue A-set Acad Amer analogue arc-curves Ayres basic set Bull cactoids called chapter characterization compact complement component conjugate connected open connected set connectedness continuous curve COROLLARY countable cut point cyclic connectivity theorem cyclic element decomposition cyclic element theory cyclicly extensible dendrite disconnected disjoint Duke Math endelements endpoints EXAMPLE FALSEHOOD Figure fine-cyclic finite flocal separation free of cut H-set hence homeomorphism interior element intersection irreducible cutting irreducible separating set limit point locally connected locally cyclicly connected maximal cyclic curve maximal with respect metric space monotone nected non-cut points non-separating point O-focally O-lfcp O-locally free open M-subelement open mapping open neighborhood open set Peano continuum Peano space plane point of X point sets pre-subelement PROOF semi-continuous semicontinuous separating point separation properties sequence simple closed curve simple cyclic chain thesis thick thin chain topological space Trans true cyclic element upper semi-continuous upper semicontinuous WHYBURN CYCLIC