A new proof of the cyclic connectivity theorem

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 at tempts 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. 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.