Continuous Collections of Continuous Curves in the Plane

The purpose of this paper is to consider the class of continuous collections of mutually exclusive compact continuous curves in the plane. Throughout this paper we shall denote by G or G with a subscript or superscript a collection of this class with the property that G with respect to its elements as points is a nondegenerate compact closed point set. G has then the significance of being both a collection of continua and a point set itself. By a continuous curve will be meant a nondegenerate locally connected compact continuum. By a continuous collection will be meant a collection which is both upper and lower semi-continuous. By a (simple) chain will be meant a finite collection Xi, x2, ■ ■ ■ , xn of open discs (i.e. interiors of simple closed curves) such that #,-•% exists if and only if \i—j\ ikl and is the closure of an open disc (i.e. a 2-cell) if it does exist. The x¿ are called links of the chain. A subchain of a chain c is a chain whose links are links of c. A chain c will be said to simply cover a set M if c* contains M and if for no proper subchain c' of c does the closure of c'* contain M. Two chains will be said to be mutually exclusive if no link of either intersects any link of the other. A collection C of sets is said to be a (closed) refinement of a collection C of sets if (the closure of) each element of C is a subset of some element of C. An emanation point of a continuum M is a point which is the common part of each pair of some three nondegenerate subcontinua of M. A hereditary continuous curve is a continuous curve each of whose nondegenerate subcontinua is a continuous curve. It is immediately clear that if G is connected, G contains uncountably many elements and that only countably many can contain triods [l]. Except for a countable number of elements, each element of G must be either an arc or a simple closed curve. We denote the elements of G which are neither arcs nor simple closed curves by gii g2, gz, • ■ ■ ■ From the hypothesis of continuity of G it follows immediately that no element of a connected G contains a 2-cell.