CONTENTS 1. Introduction......................................................................5 2. Rim-type and decompositions..........................................8 3. Defining sequences and isomorphisms..........................18 4. Embedding theorem.......................................................26 5. Construction of universal and containing spaces...........32 6. References....................................................................39
CONTENTS Introduction......................................................................................5 1. General notion of aposyndesis....................................................6 2. Relation T for special families......................................................8 3. Properties of T.............................................................................9 4. T-aposyndesis in homogeneous continua..................................11 5. Colocal connectedness and T-aposyndesis...............................13 6. Decompositions and terminal continua......................................15 7. Decompositions of homogeneous continua...............................18 8. Indecomposable continua and colocal connectedness..............20 9. Closed domains in homogeneous continua...............................22 10. Irreducible continua.................................................................23 11. Decompositions onto locally connected continua.....................25 12. Locally connected homogeneous continua..............................29 13. Atriodic homogeneous continua..............................................30 14. Arcs and pseudoarcs in homogeneous continua.....................33 15. Homogeneous continua in compact 2-manifolds......................37 16. Multicoherence and homogeneity............................................41 17. Types of aposyndesis..............................................................42 18. More about decompositions.....................................................44 19. Mappings from hereditarily indecomposable continua.............46 20. Common model........................................................................48 21. Final remarks...........................................................................52 References....................................................................................54
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A symmetric, idempotent, continuous binary operation on a space is called a mean. In this paper, we provide a criterion for the non-existence of mean on a certain class of continua which includes tree-like continua. This generalizes a result of Bell and Watson. We also prove that any hereditarily indecomposable circle-like continuum admits no mean.
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The body of this paper falls into two independent sections. The first deals with the existence of cross-sections in $F_σ$-decompositions. The second deals with the extensions of the results on accessibility in the plane.
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A space Y is called a free space if for each compactum X the set of maps with hereditarily indecomposable fibers is a dense $G_δ$-subset of C(X,Y), the space of all continuous functions of X to Y. Levin proved that the interval I and the real line ℝ are free. Krasinkiewicz independently proved that each n-dimensional manifold M (n ≥ 1) is free and the product of any space with a free space is free. He also raised a number of questions about the extent of the class of free spaces. In this paper we will answer most of those questions. We prove that each cone is free. We introduce the notion of a locally free space and prove that a locally free ANR is free. It follows that every polyhedron is free. Hence, 1-dimensional Peano continua, Menger manifolds and many hereditarily unicoherent continua are free. We also give examples that show some limits to the extent of the class of free spaces.
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CONTENTS 1. Introduction.................................................................................................................................................5 2. Partitioning Peano continua......................................................................................................................10 3. Peano continua and cross-connectedness...............................................................................................18 4. The characterization of the Menger curve.................................................................................................28 5. Extension of homeomorphisms on non-locally-separating, closed subsets of the Menger curve..............34 6. Universality and map extension theorems.................................................................................................42 7. Bibliography..............................................................................................................................................45
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The Cantor set and the set of irrational numbers are examples of 0-dimensional, totally disconnected, homogeneous spaces which admit elegant characterizations and which play a crucial role in analysis and dynamical systems. In this paper we will start the study of 1-dimensional, totally disconnected, homogeneous spaces. We will provide a characterization of such spaces and use it to show that many examples of such spaces which exist in the literature in various fields are all homeomorphic. In particular, we will show that the set of endpoints of the universal separable ℝ-tree, the set of endpoints of the Julia set of the exponential map, the set of points in Hilbert space all of whose coordinates are irrational and the set of endpoints of the Lelek fan are all homeomorphic. Moreover, we show that these spaces satisfy a topological scaling property: all non-empty open subsets and all complements of σ-compact subsets are homeomorphic.
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