A new proof of the cyclic connectivity theorem
W. L. Ayres · Bulletin of the American Mathematical Society · 1942
The cyclic connectivity theorem was first proved for the plane in 1927 by G. T. Whyburn [5].The extension of this theorem to metric space afforded some difficulty and the first proof [l] was long and tedious and complicated with convergence difficulties.A second and simpler proof appeared in 1931 [ó], but in this proof it is necessary that quite a number of properties of Peano spaces be proved in advance.This note attempts to give a new proof in which convergence troubles are encountered at just one point (step (b)) and in which just three theorems about Peano space need be known in advance: (A) Every component of an open set is open.(B) Open connected sets are arc-wise connected.(C) The space is arc-wise locally connected.Actually just two properties need to be established before cyclic connectivity can be proved, for the third theorem (C) is a simple consequence of the first two. 1 Thus the cyclic connectivity theorem may be established at the very beginning of the theory of Peano spaces and is available for use in studying other properties.