A Characterization of the Universal Curve and a Proof of Its Homogeneity

Abstract

Throughout this paper, all spaces referred to will be compact and metric. The principal result gives a characterization of the universal curve.2 The characterization involves showing, Theorem I, that certain non-planar, 1-dimensional continua (C-sets) obtainable by a wide variety of constructions are all homeomorphic to each other, and showing, Theorem II, that the universal curve is a C-set. In Theorem III, we show, as a rather straightforward corollary of the methods of Theorem I, that the universal curve is homogeneous3 and, in fact, is n-point homogeneous. The characterization of the universal curve gives a result of a type of which there are only a few known theorems. Thus, certain totally disconnected sets and 1and 2-dimensional manifolds have been characterized, as have certain elementary sums or subsets of these. The pseudoarc has been characterized by Bing [1]. This paper, in a sense, is analogous to Bing's characterization of the pseudo-arc and to some of the earlier papers on the pseudo-arc [2] and [3]. The pseudo-arc is a planar 1-dimensional hereditarily indecomposable continuum which may be characterized as the non-degenerate common part of a monotonic decreasing sequence of simple chains, each very crooked with respect to the preceding. The pseudo-arc is connected and homologically trivial, but very badly not locally connected. It is also homogeneous [2] and homeomorphic to each of its non-degenerate subcontinua [3]. The universal curve, on the other hand, is not planar, is connected and locally connected, but is very badly neither simply connected nor locally