Solution of a Problem of R. L. Wilder

Abstract

The following problem was proposed by R. L. Wilder in Concerning Simple Continuous Curves and Related Point Sets, American Journal of Mathemaatics, vol. 53 (1931), pp. 39-55: Suppose that M is a connected im Aleinen set which is the sum of two sets M1 and 3M2 such that each is irreducibly connected from the point A to the point B and such that the common part of M1 and M2 is A and B. Is M necessarily a simple closed curve? Wilder showed that M is a simple closed curve if it is a locally compact continuum. This note gives an example to show that M need not be a simple closed curve if this extra requirement is not placed on it. We shall prove the following for the plane: Suppose that S is a square plus its interior. There exists a collection G of point sets filling up S such that each element of G is irreducibly connected from a point A to a point B, the common part of two elements of G is A + B, and the sum of two elements of G is a locally connected subset of S which is dense in S. Let A and B be opposite vertices of S and let E and F be the other vertices. Denote by [X] the collection of all straight line intervals EP where P is a point of AF + FB(A + B). For each element X of [X], let L (X) denote the collection of all straight line intervals which have their end points on the boundary of S and which are parallel to X. We note that with respect to its elements, each L (X) is an open arc from A to B. Let [L] be the set of all such collections L(X). There exists a well ordered sequence W(L) such that the elements of W(L) are the eloments of [L] and such that if L is an element of [L], then the set of all elements of [L] which precede L in W (L) has a power less than that of [L]. We note that [L] has the power of the continuum. We shall denote the element of W(L) whose ordinal number is a by La. Let W(S) be a well ordering of the points of S. Denote the collection of all subcontinua of S by [C]. Points as well as nondegenerate continua are included in [C]. Let TW (C) be a well ordered sequence such that the elements of W(C) are the elements of [C] and such that if C is a subcontinuum of S, then the collection of all elements of [Cl