On Several Families of Plane Curves with an Application to Differential Equations

Abstract

1. Let D be a domain bounded by a closed Jordan curve J. Let us assume that a family F of simple continuous curves is defined in the domain, with the conditions that 1: Through every point of D and of its boundary passes one and only one curve of the family. (The curve through the point P will be designated by C(P).) 2: No curve of the family has an end point in the interior of D. THEOREM. The curve C(P), regarded as a function of P, is continuous; that is to say, if P. is a sequence of points converging to the point P, the curves C(P(F) converge uniformly to the curve C(P). PROOF. Let us suppose our theorem false. Then, there exists a positive number e and a sequence Pni contained in the sequence PR, such that each curve C(Pni) has a point pi whose distance from the curve C(P) is greater than C. Let p be a limit point of the set pi. This point p is not on the curve C(P). From the sequence C(Pi) we can extract a new sequence such that the corresponding points pi converge to p. Then, successively extracting new sequences of curves from this sequence, each sequence contained in the preceding, and using the well known method of the diagonal, we can obtain a sequence of curves C* which possesses the following property: If E is the set of the points x such that every neighborhood of x has points in common with an infinite number of the curves C*, then, if 5 is a given neighborhood of x, there exists a number N such that every curve C*, with n > N, has points in S. The set E is obviously a bounded continuum joining P and p. Now, let us divide the points of E into disjoint sets E, in such a manner that two points of E belong to E, if and only if they belong to a same curve of the family F. Every set E,, being the intersection of two closed sets (the set E and a curve of the family F), is closed. As is well known,' a bounded continuum cannot be divided into a finite number (greater than 1), or into a countable number of closed disjoint sets. Therefore, if we prove that the sets E. are not uncountable, it will follow that there exists only one set E,. Then the set E would be an arc of a curve C, of the family F, joining P and p, and consequently distinct from the curve C(P) which does not contain the point p. But this is impossible, since C(P) is the only curve of the family F that passes through P.