A Characterization of a Semi-locally-connected Plane Continuüm

In 1908 Schoenflies characterized a continuous curve (in the plane) by means of its complement. In view of more recent work, the kernel of his characterization may be stated as follows : In order that a plane, bounded, cyclic continuum be a continuous curve it is necessary and sufficient that (1) each of its complementary domains be simple and (2) the collection of its complementary domains be contracting. In his paper on semi-locally-connected sets, G. T. Whyburn pointed out many similarities between semi-locally-connected sets and continuous curves. In particular (it is an immediate consequence of one of his theorems) every complementary domain of a plane, bounded, cyclic semi-locally-connected continuum is simple. But since such a continuum need not be a continuous curve, the collection of its complementary domains need not be contracting. I t is the purpose of this paper to point out what characteristic property this collection of complementary domains does possess : namely, the collection contains no folded subcollection. This property, in conjunction with (1) above, characterizes a plane, bounded, cyclic semi-locally-connected continuum. In fact it is shown that a plane, bounded continuum, whether cyclic or not, is semi-locally-connected if and only if its complement is non-folded. These theorems should prove useful when constructing examples of semi-locally-connected continua which also have certain other properties.