On embedding cones over circularly chainable continua

Abstract

It is known that every chainable continuum (i.e., every 1-cell-like continuum) can be embedded in the plane [1]. Every 2-cell-like continuum can be embedded in E4 [5]. It is natural to ask whether every 2-cell-like continuum can be embedded in El. Fearnley [3] has stated that the cone over a dyadic solenoid cannot be embedded in Et, but no proof has appeared. In this note we show that the cone over a nonplanar circularly chainable continuum cannot be embedded in E3. This provides a large class of 2-cell-like continua not embeddable in E3. A definition of what it means to say that a continuum is P-like may be found in [6]. For any continuum M, the cone over M is the decomposition space C(M) of the upper semicontinuous decomposition of MX [0, 11 whose only nondegenerate element is MX { 1). We call the point MX { 1 1 of C(M) the vertex of C(M) and the set of points { (m, 0), m in M}, we call the base of C(M). It is easy to see that if P is a continuum and M is P-like then C(M) is C(P)-like. Therefore every cone over a circularly chainable continuum is 2-cell-like.