Semi-local-connectedness and cut points in metric continua

Abstract:

In the first section of this paper, the notion of a space being rational at a point is generalized to what is here called quasi-rational at a point. It is shown that a compact metric continuum which is quasi-rational at each point of a dense subset of an open set is both connected im kleinen and semi-locally-connected on a dense subset of that open set. In the second section a ${G_\delta }$ set is constructed such that every point in the ${G_\delta }$ at which the space is not semi-locally-connected is a cut point. A condition is given for this ${G_\delta }$ set to be dense. This condition, in addition to requiring that the space be not semi-locally-connected at any point of a dense ${G_\delta }$ set gives a sufficient condition for the space to contain a ${G_\delta }$ set of cut points. The condition generalizes that given by Grace.