On the distributivity and simple connectivity of plane topological lattices

Abstract

D. Edmundson [1] has recently constructed a counter-example to the conjecture by A. D. Wallace that every compact connected topological lattice is distributive. He discovered a subset of Euclidean three-space which, with the appropriate lattice operations, forms a nonmodular compact connected topological lattice. The main result of this paper states that if a locally compact connected subset of Euclidean two-space admits a pair of continuous lattice operations then the lattice is distributive. 1. Preliminaries. We recall that a topological lattice is a Hausdorff space, L, together with a pair of continuous furnctions A: L XL-*yL and V: L XL- yL which satisfy the usual conditions stipulated for a lattice. As is usual we denote A(x, y) byxAy and V(x, y) byxVy. Unless explicitly stated to the contrary, we reserve the symbols 0 and 1 to denote the unique minimal and maximal elements of a lattice, whenever they exist. A subset, C, of L is a chain if xAy=x or y for any pair of elements x and y in C. If a and b are elements of L with a ? b then C is said to be a chain from a to b provided C is a chain contained in a V(bAL) and containing both a and b. If A is a subset of L, we deiiote by A*, A' and F(A) =A *\A' the closure, interior and boundary of A respectively. Throughout this paper, R2 will denote the Cartesian plane with the usual topology, SI will deniote the unit circle and I will denote the closed real number interval [(, 1]. A subset of R2 is a simple closed curve if it is a homeomorph of S1. We reccll that Jordan's theorem [2] states that any simple closed curve cuts R2 into exactly two components, one bounded and the other unbounded. If C is a simple closed curve, we denote by B(C) the bounded component of R2\C. If X is a topological space and A is a subset of X, we denote by IIn(X, A) the n-dimensional Alexander-Kolmogoroff cohomology group of X modulo A with coefficients in some fixed nontrivial additive abelian group. In this paper we utilize two dimension functions, namely codimension (cd) and inductive dimension (ind). For the definitions and essential theorems relating to these dimension functions the reader is referred to the work of H. Cohen [3]. It is a pleasure to acknowledge the advice and suggestions of A. D. Wallace in the preparation of this paper.