Theory of relative retracts
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- Kod: 5991
- Producent: Wydawnictwo Politechniki Koszalińskiej
- Autor: Mirosław Ślosarski
- Rok wydania: 2021
- ISBN: 978-83-7365-553-9
- Liczba stron: 155
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Opis
The theory of relative retracts is a generalization and supplementation of K. Borsuk's theory of retracts. For the construction of relative retracts we use the properties of some notions of K. Borsuk's theory of shape such as: the trivial shape and movability of metric spaces. For studying their properties we use new topological tools such as: relative homotopy, relative contractibility and relative extension of continuous mappings. Relative retracts have many diferent applications. In this monograph they are applied to the fixed point theory of compact and non-compact mappings, to the theory of coincidence and to the study of local and global properties of metric spaces.
Spis treści
1. Introduction / 7
2. Preliminaries / 11
2.1 Generalized Lefschetz number / 11
2.2 The Čech homology functor with compact carriers / 12
2.3 Proper maps / 12
2.4 Properties of ANR-s (AR-s) / 14
2.5 Compact spaces in the Hilbert cube / 16
2.6 Multivalued maps / 18
2.7 Topological spaces / 19
3. Multifunctions / 23
3.1 Multimorphisms / 23
3.2 Homotopy of multimorphisms / 30
4. Relative homotopy / 35
4.1 Families of sets of D type / 35
4.2 Relative homotopy / 37
4.3 Relative contractibility / 40
5. Relative dominations / 45
5.1 Relations in the set DC(X,Y) / 45
5.2 D-retracts / 47
5.3 Relative domination / 48
5.4 Relative homotopy domination / 50
5.5 Deformation D-retracts / 51
5.6 Deformation V-retracts / 54
6. Relative retracts / 57
6.1 Relative retracts / 57
6.2 Properties of relative retract / 59
6.3 MOV-retracts / 62
6.4 CELL-retracts / 70
6.5 Examples of CELL-retracts / 72
6.6 Other relative retracts / 77
7. Relative extension / 83
7.1 Relative extension of continuous maps / 83
7.2 Relative extension of homotopy / 85
8. Multi-domination / 87
8.1 Multi-dimination of metric spaces / 87
8.2 Properties of multi-domination / 91
9. Fixed points and the coincidence points / 97
9.1 Fixed points of compact maps / 97
9.2 Fixed points of non-compact maps / 98
9.3 Coincidence points / 100
10. Properties of metrizable spaces / 103
10.1 Characterization of ANR-s / 103
10.2 Acyclic sets / 106
10.3 Relative types of homotopy / 109
11. Approximative relative retracts / 111
11.1 Approximative relative retracts / 111
11.2 Properties of approximative relative retracts / 114
11.3 Examples of approximative relative retracts / 116
11.4 Application of approximative relative retracts / 118
12. Multivalued maps / 123
12.1 Elementary extension of multivalued maps / 123
12.2 CELL-admissible maps / 131
13. Relative retracts in topological spaces / 141
13.1 Families of D type sets / 141
13.2 Spaces of N ESR(ESR) type / 144
13.3 Fixed points of compact maps / 148
13.4 Fixed points of noncompact maps / 149
References / 151
Index / 155
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