Concerning the Cut Points of Continua*
نویسنده
چکیده
In this paper propositions will be established concerning the collection of all the cut points of a given planef continuum. It will be shown that there does not exist an uncountable collection of mutually exclusive subcontinua of a given continuum M each of which contains at least one cut point of M. With the aid of this fundamental theorem it is shown, among other things, that the set of all the cut points of a continuum M which are irregular points of M is necessarily countable and, indeed, that all save a countable number of the cut points of any continuum M are points of Menger order two of M.
منابع مشابه
Two - Way Continuous Curves
A continuous curve M will be said to be a two-way continuous curve, or to be "two-way continuous," provided it is true that between every two points of M there exist in M at least two arcs neither of which is a subset of the other. A point P of a continuum M is a cut point of M provided it is true that the point set M—P is not connected. Every point of a continuum M which is not a cut point of ...
متن کاملFrom continua to R-trees
We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X . AMS Classification 54F15,54F05,20F65,20E08
متن کاملConcerning Continua in a Separable Space Which Do Not Cross*
In working with collections of continua it is sometimes useful to know something of the character of the point set consisting of all the points common to two or more members of the collection. Also it is of advantage to know conditions under which we may subtract a countable number of continua from the collection and have left a collection of mutually exclusive continua. The theorem which we sh...
متن کاملConnectedness Properties of Limit Sets
We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.
متن کاملEnds of Cusp-Uniform Groups of Locally Connected Continua - I
Due to works by Bestvina-Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to gener...
متن کامل