An example of a nonarcwise connected rim-countable locally connected continuum

1999 ◽  
Vol 65 (5) ◽  
pp. 553-559
Author(s):  
S. A. Drozdovskii
Keyword(s):  
Locally Connected Continuum

A Note On Arc-Preserving Functions For Manifolds1

1967 ◽  
Vol 10 (4) ◽  
pp. 597-598
Author(s):  
H. J. Charlton
Keyword(s):  
Compact Manifolds ◽  
Locally Connected Continuum

Hall and Puckett [2] have shown that an arc - preserving function defined on a locally connected continuum having no local separating points is a homeomorphism if its total image is not an arc or point. This note shows that their results can be extended to non-compact manifolds.

ENDS OF CUSP-UNIFORM GROUPS OF LOCALLY CONNECTED CONTINUA — I

2005 ◽  
Vol 15 (04) ◽  
pp. 765-798 ◽  
Author(s):  
DAN P. GURALNIK
Keyword(s):  
Hyperbolic Group ◽  
Infinite Group ◽  
Graph Of Groups ◽  
Locally Finite ◽  
Cut Point ◽  
Word Hyperbolic Group ◽  
Complete Knowledge ◽  
Simplicial Tree ◽  
Locally Connected Continuum ◽  
Uniform Group

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 generalize this theory to cusp-uniform groups in the sense of Tukia. A Peano space X is cut-rigid, if X has no cut point, no points of infinite valence and no cut pairs consisting of bivalent points. We prove: Theorem. Suppose X is a cut-rigid space admitting a cusp-uniform action by an infinite group. If X contains a minimal cut triple of bivalent points, then there exists a simplicial tree T, canonically associated with X, and a canonical simplicial action of Homeo(X) on T such that any infinite cusp-uniform group G of X acts cofinitely on T, with finite edge stabilizers. In particular, if X is such that T is locally finite, then any cusp-uniform group G of X is virtually free.

Separators in Continuous Images of Ordered Continua and Hereditarily Locally Connected Continua

1993 ◽  
Vol 36 (2) ◽  
pp. 154-163 ◽  
Author(s):  
J. Grispolakis ◽  
J. Nikiel ◽  
J. N. Simone ◽  
E. D. Tymchatyn
Keyword(s):  
Open Subset ◽  
Hausdorff Space ◽  
Continuous Image ◽  
Open Problems ◽  
Connected Open Subset ◽  
Locally Connected Continuum ◽  
Continuous Images

AbstractLet X be a Hausdorff space which is the continuous image of an ordered continuum. We prove that every irreducible separator of X is metrizable. This is a far reaching extension of the 1967 theorem of S. Mardešić which asserts that X has a basis of open sets with metrizable boundaries. Our first result is then used to show that, in particular, if Y is an hereditarily locally connected continuum, then for subsets of Y quasi-components coincide with components, and that the boundary of each connected open subset of Y is accessible by ordered continua. These results answer open problems in the literature due to the fourth and third authors, respectively.

scholarly journals A property of hereditarily locally connected continua related to arcwise accessibility

1978 ◽  
Vol 25 (1) ◽  
pp. 35-40 ◽  
Author(s):  
Joseph N. Simone
Keyword(s):  
Open Subset ◽  
Hausdorff Space ◽  
Open Set ◽  
Connected Open Subset ◽  
Locally Connected Continuum

AbstractA continuum (that is, a compact connected Hausdorff space) is hereditarily locally connected if each of its subcontinua is locally connected. It is shown that a continuum X is hereditarily locally connected if and only if for each connected open set U in X and each point p in the boundary of U, U ∪ {p} is locally connected. This result is used to prove that if X is an hereditarily locally connected continuum, U is a connected open subset of X, p is an element of the boundary of U and X is first countable at p, then p is arcwise accessible from U.

scholarly journals Elusive Examples of Non-Metrizable Continua which Admit a Whitney Map

2007 ◽  
Vol 14 (4) ◽  
pp. 711-719
Author(s):  
Ivan Lončar
Keyword(s):  
Locally Connected Continuum

Abstract The main purpose of this paper is to prove that the class of near locally connected continua contains no non-metrizable continuum 𝑋 which admits a Whitney map for 𝐶(𝑋). In particular, each near locally connected continuum 𝑋 which admits a Whitney map for 𝐶(𝑋) is metrizable.

scholarly journals A note on periodic points and recurrent points of maps of dendrites

1995 ◽  
Vol 51 (3) ◽  
pp. 459-461 ◽  
Author(s):  
Hisao Kato
Keyword(s):  
Simple Closed Curve ◽  
Periodic Points ◽  
Unit Interval ◽  
Closed Curve ◽  
Locally Connected Continuum

Let f: X → X be a map of a continuum X. Let P(f) denote the set of all periodic points of f and R(f) denote the set of all recurrent points of f. In [2], Coven and Hedlund proved that if f: I → I is a map of the unit interval I = [0, 1], then CI(P(f)) = CI(R(f)). In [7], Ye generalised this result to maps of a tree. It is natural to ask whether the result generalises to maps of a dendrite. (A dendrite is a locally connected continuum which contains no simple closed curve.) The aim of this paper is to show that the answer is negative.

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