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Containing spaces for planar rational compacta

100%
Mayer J. C., Tymchatyn E. D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction.............................................................................5 2. Ordered scattered spaces......................................................6  2.1. Topological type..................................................................6  2.2. Ordered spaces..................................................................6  2.3. Rim-type.............................................................................9  2.4. Disk partitions.....................................................................9 3. Partition theorems.................................................................12  3.1. Partition Pullback Theorem...............................................13  3.2. Defining sequences of partitions.......................................13  3.3. a-Circlelacing lemmas.......................................................14  3.4. Brick Pulverizing Lemma...................................................15  3.5. Construction of a containing space...................................20 4. Embedding Theorem.............................................................21 5. Universal spaces...................................................................24  5.1. Question...........................................................................25  5.2. The Sierpiński curve.........................................................25  5.3. A conjectured universal planar rational space...................26  5.4. A conjectured universal planar space of rim-type ≤ a........26 References................................................................................26
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Universal rational spaces

100%
Mayer J. C., Tymchatyn E. D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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
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Continuous mappings on continua II

100%
Maćkowiak T., Tymchatyn E. D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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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Continua which admit no mean

91%
Kawamura K., Tymchatyn E. D.
Colloquium Mathematicum
|
1996
|
tom 71
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nr 1
97-105
EN
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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Composant-like decompositions

91%
Dębski W., Tymchatyn E. D.
Fundamenta Mathematicae
|
1991-1992
|
tom 140
|
nr 1
69-78
EN
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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Free spaces

91%
Song J. W., Tymchatyn E. D.
Fundamenta Mathematicae
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2000
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tom 163
|
nr 3
229-239
EN
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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Irreducible continua with degenerate end-tranches and arcwise accessibility in hyperspaces

91%
Grispolakis J., Tymchatyn E. D.
Fundamenta Mathematicae
|
1980
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tom 110
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nr 2
117-130
8
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The Menger curve Characterization and extension of homeomorphisms of non-locally-separating closed subsets

83%
Mayer J. C., Oversteegen L. G., Tymchatyn E. D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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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On homogeneous totally disconnected 1-dimensional spaces

74%
Kawamura K., Oversteegen L. G., Tymchatyn E. D.
Fundamenta Mathematicae
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1996
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tom 150
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nr 2
97-112
EN
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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Inaccessibility, essential maps, and shape theory

73%
Brechner B. L., Mayer J. C., Tymchatyn E. D.
Fundamenta Mathematicae
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1989
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tom 132
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nr 1
1-23
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The theorem of Miss Mullikin-Mazurkiewicz-van East for unicoherent Peano spaces

73%
Tymchatyn E. D., Hunt J. H. V.
Fundamenta Mathematicae
|
1972-1973
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tom 77
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nr 3
285-287
12
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On three problems of Franklin and Wallace concerning partially ordered spaces

73%
Tymchatyn E. D., Ward, Jr. L. E.
Colloquium Mathematicum
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1969
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tom 20
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nr 2
229-236
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Sets of end-points and ramification points in dendroids

73%
Nikiel J., Tymchatyn E. D.
Fundamenta Mathematicae
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1991
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tom 138
|
nr 2
139-146
14
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Some properties of open and related mappings

72%
Maćkowiak T., Tymchatyn E. D.
Colloquium Mathematicum
|
1984-1985
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tom 49
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nr 2
175-194
15
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On hereditarily indecomposable compacta

72%
Oversteegen L. G., Tymchatyn E. D.
Banach Center Publications
|
1986
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tom 18
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nr 1
407-417
16
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Continua whose connected subsets are arcwise connected

72%
Tymchatyn E. D.
Colloquium Mathematicum
|
1971-1972
|
tom 24
|
nr 2
169-174
17
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Weakly confluent mappings and a classification of continua

72%
Tymchatyn E. D.
Colloquium Mathematicum
|
1976
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tom 36
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nr 2
229-233
18
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Some order theoretic characterizations of the 3-cell

72%
Tymchatyn E. D.
Colloquium Mathematicum
|
1971-1972
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tom 24
|
nr 2
195-203
19
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Hereditarily σ-connected continua

64%
Grispolakis J., Tymchatyn E. D.
Colloquium Mathematicum
|
1979
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tom 41
|
nr 1
57-65
20
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On span and weakly chainable continua

64%
Oversteegen L. G., Tymchatyn E. D.
Fundamenta Mathematicae
|
1984
|
tom 122
|
nr 2
159-174
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