Irreducible continua and generalization of hereditarily unicoherent continua by means of membranes

Abstract

In [2] we defined an irreducible B(J)-cartesian membrane, and used this to obtain a characterization of an n-sphere by generalizing the definition of simple closed curve given by Theorem 1.2 below. There B(J) is a class A(n) of (n−1)-spheres, but here it is a class of mutually homeomorphic continua. In Theorem 1.1 we give a definition of hereditarily unicoherent continua and generalize this in Section 3 by means of B(J)-cartesian membranes. To do this we paraphrase by a translation some of Wilder's work in [7]. In his Unified Topology [8: p. 674] he gives a principle: “The connectedness of a domain is a special case of the bounding properties of its i-cycles”. We substitute the element J of B(J) for the i-cycle and for “bound” we substitute that “J membrane-bases an irreducible B(J)-cartesian membrane. The very nature of an i-cycle seems to limit the complexity of the point set studied, although the restriction to “nice” manifolds is due partly to the difficulty of the subject matter treated. There are similar difficulties here, but also advantages, in the very general set-theoretic approach by means of B(J)-membranes.