ABSTRACT. Using a synthesis of techniques involving topological groups, inverse limits, geometric... more ABSTRACT. Using a synthesis of techniques involving topological groups, inverse limits, geometric topology, compactifications, and orbit spaces of involutions on continua, a continuum with the properties described in the title is constructed. Introduction. The fixed-point property has been the subject of intense investigation which cannot be adequately summarized here. The interested reader is referred to Bing's survey article [ 1] for historical and general background. In the last few years, important advances have been made. Notably, L. Mohler [7] has shown that uniquely arcwise connected continua have the fixed point property for
David P. Bellamy asked, If T is continuous for a continuum S, is K also continuous for S? We pres... more David P. Bellamy asked, If T is continuous for a continuum S, is K also continuous for S? We present a class of continua for which the answer to this question is positive.
Proceedings of the American Mathematical Society, 1992
Using the universal cover of the Möbius band, we construct an example as described in the title w... more Using the universal cover of the Möbius band, we construct an example as described in the title which answers a question of K. Kuperberg.
Proceedings of the American Mathematical Society, 1971
In this paper we prove that if r is a filterbase of closed subsets of a compact Hausdorff space t... more In this paper we prove that if r is a filterbase of closed subsets of a compact Hausdorff space then r(flr) = n {r(G)|GGr}, where T(A) denotes the set of those points for which every neighborhood which is a continuum intersects A nonvoidly.
Proceedings of the American Mathematical Society, 1971
We show that every compact metric continuum is a continuous image of some indecomposable compact ... more We show that every compact metric continuum is a continuous image of some indecomposable compact metric continuum.
This article concerns the properties of certain spaces which can occur as Stone-Cech remainders o... more This article concerns the properties of certain spaces which can occur as Stone-Cech remainders of locally compact Hausdorff spaces. I want to thank B. Diamond for two very useful conversations on this topic, and for calling to my attention Lemma 1 which made this work possible. A continuum is a compact, connected Hausdorff space. Let Y be a continuum. Y is irreducible between the points a,b E Y if no proper subcontinuum of Y contains both of them. This will be denoted by Y = [a,b], with the under standing that if a,b E R, the usual meaning applies. Y is connected im Kleinen at p E Y provided every neighborhood of p contains a neighborhood of p which is connected and closed in Y. Y is indecomposable if it is not the union of two of its proper subcontinua, or equivalently if every proper subcontinuum of Y is nowhere dense. If p E Y, the
The theorem proven here is that every compact metric continuum is a continuous image of some here... more The theorem proven here is that every compact metric continuum is a continuous image of some hereditarily indecomposable metric continuum.
The main result of this article is: Theorem. Every homogeneous locally conical connected separabl... more The main result of this article is: Theorem. Every homogeneous locally conical connected separable metric space that is not a 1-manifold is strongly n-homogeneous for each n ≥ 2. Furthermore, every homogeneous locally conical separable metric space is countable dense homogeneous. This theorem has the following two consequences. Corollary 1. If X is a homogeneous compact suspension, then X is an absolute suspension (i.e., for any two distinct points p and q of X, there is a homeomorphism from X to a suspension that maps p and q to the suspension points). Corollary 2. If there exists a locally conical counterexample X to the Bing-Borsuk Conjecture (i.e., X is a locally conical homogeneous Euclidean neighborhood retract that is not a manifold), then each component of X is strongly n-homogeneous for all n ≥ 2 and X is countable dense homogeneous.
ABSTRACT. Using a synthesis of techniques involving topological groups, inverse limits, geometric... more ABSTRACT. Using a synthesis of techniques involving topological groups, inverse limits, geometric topology, compactifications, and orbit spaces of involutions on continua, a continuum with the properties described in the title is constructed. Introduction. The fixed-point property has been the subject of intense investigation which cannot be adequately summarized here. The interested reader is referred to Bing's survey article [ 1] for historical and general background. In the last few years, important advances have been made. Notably, L. Mohler [7] has shown that uniquely arcwise connected continua have the fixed point property for
David P. Bellamy asked, If T is continuous for a continuum S, is K also continuous for S? We pres... more David P. Bellamy asked, If T is continuous for a continuum S, is K also continuous for S? We present a class of continua for which the answer to this question is positive.
Proceedings of the American Mathematical Society, 1992
Using the universal cover of the Möbius band, we construct an example as described in the title w... more Using the universal cover of the Möbius band, we construct an example as described in the title which answers a question of K. Kuperberg.
Proceedings of the American Mathematical Society, 1971
In this paper we prove that if r is a filterbase of closed subsets of a compact Hausdorff space t... more In this paper we prove that if r is a filterbase of closed subsets of a compact Hausdorff space then r(flr) = n {r(G)|GGr}, where T(A) denotes the set of those points for which every neighborhood which is a continuum intersects A nonvoidly.
Proceedings of the American Mathematical Society, 1971
We show that every compact metric continuum is a continuous image of some indecomposable compact ... more We show that every compact metric continuum is a continuous image of some indecomposable compact metric continuum.
This article concerns the properties of certain spaces which can occur as Stone-Cech remainders o... more This article concerns the properties of certain spaces which can occur as Stone-Cech remainders of locally compact Hausdorff spaces. I want to thank B. Diamond for two very useful conversations on this topic, and for calling to my attention Lemma 1 which made this work possible. A continuum is a compact, connected Hausdorff space. Let Y be a continuum. Y is irreducible between the points a,b E Y if no proper subcontinuum of Y contains both of them. This will be denoted by Y = [a,b], with the under standing that if a,b E R, the usual meaning applies. Y is connected im Kleinen at p E Y provided every neighborhood of p contains a neighborhood of p which is connected and closed in Y. Y is indecomposable if it is not the union of two of its proper subcontinua, or equivalently if every proper subcontinuum of Y is nowhere dense. If p E Y, the
The theorem proven here is that every compact metric continuum is a continuous image of some here... more The theorem proven here is that every compact metric continuum is a continuous image of some hereditarily indecomposable metric continuum.
The main result of this article is: Theorem. Every homogeneous locally conical connected separabl... more The main result of this article is: Theorem. Every homogeneous locally conical connected separable metric space that is not a 1-manifold is strongly n-homogeneous for each n ≥ 2. Furthermore, every homogeneous locally conical separable metric space is countable dense homogeneous. This theorem has the following two consequences. Corollary 1. If X is a homogeneous compact suspension, then X is an absolute suspension (i.e., for any two distinct points p and q of X, there is a homeomorphism from X to a suspension that maps p and q to the suspension points). Corollary 2. If there exists a locally conical counterexample X to the Bing-Borsuk Conjecture (i.e., X is a locally conical homogeneous Euclidean neighborhood retract that is not a manifold), then each component of X is strongly n-homogeneous for all n ≥ 2 and X is countable dense homogeneous.
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