A note on continuous collections of continuous curves filling up a continuous curve in the plane

Abstract

This theorem provides a complement to Theorem VI of [1] which states that if G is a continuous collection of nondegenerate compact continuous curves in the plane which is a compact continuum with respect to its elements as points, then G is a hereditary continuous curve such that the closure of its set of emanation points is totally disconnected. Some notation similar to that of [1 ] will be used. A simple chain is a finite collection x1, x2, * * *, xn of open discs (interiors of simple closed curves) such that xi x; exists if and only if I i-il < 1 and is a closed disc (a simple closed curve plus its interior) if it does exist. A (simple) chain C is said to simply cover a set M if C* contains M but for no proper subchain C' of C does the closure of C'* contain M.' Two chains, C and C', are said to be mutually exclusive if C* and C'* are mutually exclusive. The chain C is said to be a closed refinement of the chain C' provided that each link of C' contains the closure of some link of C and the closure of each link of C is contained in some link of C'. PROOF OF THEOREM. We note first that only a countable number of the elements of G contain triods. (See [5].) Denote the elements of some subset of G containing these elements by gl, g2, * C C . Each of the remaining elements of G is an arc or a simple closed curve. If G is not an arc or a simple closed curve with respect to its elements as points, it has an emanation element. Let H denote the closure of the set of emanation elements of G. It has been noted that, by Theorem VI of [1], H is totally disconnected. Let T be an arc of elements of G and suppose that the non-endelement t of T is an isolated element of T H. There is a subarc T1 of