Concerning non-dense plane continua

Karl Menger has shown that a necessary and sufficient condition that a plane continuum M contains no domain is that for each point P of M and each positive number e there exists a positive simple closed curve J of diameter less than e which encloses P, and such that the common part of M and J is totally disconnected. Paul Urysohn obtained the slightly weaker result that for each point P of M and each positive number e there exists a totally disconnected subset T of M such that M-T is the sum of two mutually separated sets H and K, such that H contains P and is of diameter less than e. In the present paper it is shown that if M is a continuum which contains no domain then there exists a set G of simple closed curves filling the whole plane and indeed topologically equivalent to the set of all polygons, such that the common part of M and any simple closed curve of the set G is vacuous or is totally disconnected. Additional results are obtained for the special case where the continuum M is a continuous curve