Certain problems related to the cutting of a simply connected plane region by a continuum

Abstract

1. If R is a simply connected plane region whose frontier F is bounded and C is a simple arc lying in R save for its end points, it is well known that C enjoys the following properties: (a) R R C consists of two simply connected regions R1 and R2; (b) C is a part of the frontiers of both R1 and R2; (c) F+C is the union of the frontiers of R1 and R2. Nothing could be simpler than this theorem, but when one desires to use it, certain questions arise. For instance, what precisely are the frontiers of R1 and R2? If F is the union of two continua H and K, can C be so drawn that its ends lie on two components of H. K? If so, will the frontiers of the two regions be C+H and C+K? These questions are readily answered in the case that F is a simple closed curve, but the answers in this case do not always remain correct when F is more complicated. In particular, a negative answer to the second question suggests the substitution of a more general continuum for C and this in turn leads to a re-examination of the truth of the resulting extension of the theorem. Such an extension has been made elsewheret and it is intended in the present article to examine further these questions and certain related topics. The first and third of the questions suggested above are discussed in ??3-6. The remainder of the paper deals with the situation arising when the arc C is replaced by a special kind of irreducible continuum and with related questions of accessibility. In particular, it is shown that under certain broad conditions the second and third of the above questions can be answered in the affirmative if the arc C is replaced by a slightly more general continuum. 2. Notation. In addition to the common notation of the aggregate theory, the following will be used. The whole plane is denoted throughout by Z. If F is a continuum, a component of Z F whose frontier is F is called a principal component; a component of Z F whose frontier is a proper part of F is called secondary. A