Dendritic compactifications of certain dendritic spaces

Abstract

A dendritic space is a connected space in which every two points are separated by a third point. In this paper we describe a very natural method for obtaining a dendritic compactification of any connected space for which a dendritic compactification exists. The method is an extension of the familiar process of compactifying E 1 by adjoining-oo and + oo. In what follows, an arc is a Hausdorff continuum with only two noncut points. A ray is an arc minus one of its noncut points. The space X is semi-locally connected at the point p if each open set containing p contains an open set V containing p such that X-V has at most finitely many components. LEMMA. If the space X is arcwise connected but is not semilocally connected at the point p, then there exists an open set U containing p such that if V is an open set containing p and lying in U, then X-V has infinitely many components that intersect both V and X-U.