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, set is constructed such that every point in the GQ at which the space is not semi-locally-connected is a cut point. A condition is given for this G, set to be dense. This condition, in addition to requiring that the space be not semi-locally-connected at any point of a dense G5 set gives a sufficient condition for the space to contain a G, set of cut points. The condition generalizes that given by Grace. 1. Throughout this paper M will be taken to be a compact metric continuum. Many of the lemmas, however, can be proven with less hypotheses. Lemma 2, for example, requires only that the sets P, (defined below) be subcontinua of M. Compact Hausdorff is sufficient for this to happen [4]. Let x, y, and z be points of M (not necessarily distinct). The point x cuts between y and z in M when every subcontinuum of M which contains both y and z must also contain x. The point x is a cut point of M when x cuts between two points distinct from x. M is said to be aposyndetic (semi-locally-connected) at x wvith respect to y if and only if there is a subcontinuum of M with x (y) in its interior that does not contain y (x). M is aposyndetic (semi-locally-connected) at x when it is aposyndetic (semi-locally-connected) at x with respect to every other point. Finally, M is connected im kleinen at x when each neighborhood of x contains a closed neighborhood of x which is also connected. One should note that when M is connected im kleinen at a point, it is also aposyndetic at that point. For x E M, P, denotes {y E M I M is not aposyndetic at y with respect to x}, and for T c M, PT denotes n {H I T c HO and H is a Received by the editors May 20, 1970. AMS 1969 subject classifications. Primary 5455.