The cyclic connectivity of plane continua
F. Burton Jones · Pacific Journal of Mathematics · 1961
Suppose that p and q are distinct points of the compact plane continuum M. If no point separates p from q in M and M is locally connected, then it is known [5] that M contains a simple closed curve which contains both p and q.But in the absence of local connectivity such a simple closed curve may fail to exist.Even if no point cuts 1 p from q in M, there does not necessarily exist in M a simple closed curve which contains both p and q.For example, no point of the continuum C indicated in Figure 1 cuts p from q in C, but C contains no simple closed curve whatsoever.However, if M is the continuum obtained by adding to C either of its complementary domains, there does exist in M a simple closed curve which contains both p and q.Here M fails to separate the plane and this is indicative of the general situation.Fig. 1 LEMMA.If p is a point of the compact subcontinuum M f of the plane S and U is a nondegenerate compact continuum containing p