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     Monografie Matematyczne    

Tom 59

Theory of shape

Karol Borsuk

Spis treści

CHAPTER I RPRELIMINARIES 

 

CHAPTER II WEAK AND STRONG FUNDAMENTAL SEQUENCES 

 

CHAPTER III W-SHAPES ANS S-SHAPES 

§1 W-domination and S-domination rel. M,N. W-equivalence and S-equivalence rel. M,N
§2 Pseudo-domination and pseudo-equivalence
§3 Case of closed pairs
§4 Case of ANR(M)-spaces
§5 The concept of W-shape and of S-shape
§6 W-invariants and S-invariants
§7 Cartesian product of W-shapes
§8 An example
§9 Facotrs of W-shapes
§10 Suspensions of W-shapes and of S-shapes
§11 S-shape of an S-deformation retract
§12 S-shape of some of some decomposition spaces
§13 Similar decompositions of Euclidean spaces
§14 W-dimension and S-dimension
§15 W-triviality and S-triviality of one pair with respect to another pair
§16 Cohomotopical n-admissibility as an S-invariant 

CHAPTER IV W-SHAPES ANS S-SHAPES OF POINTED SPACES 

§1 Pointed W-shapes and pointed S-shapes
§2 Approximative maps of pointed compacta
§3 Approximative maps of pointed spheres
§4 Fundamental group
§5 Homomorphisms of fundamental groups induced by pointed W-classes
§6 Addition of pointed W-shapes and of pointed S-shapes
§7 Multioplication of pointed W-shapes
§8 Approximative n-connectedness
§9 Approximative contractibility
§10 The case of ANR(M)-spaces 

CHAPTER V MOVABLE SPACES 

§1 Movable subsets of a space
§2 Examples of movable and of non-movable spaces
§3 Suspension of movable spaces
§4 Cartesian product of movable spaces
§5 Contractibibity and W-contractibility of a space to its closed subset
§6 Some examples of movable spaces
§7 Components of movable compacta
§8 Movable pointed spaces
§9 Addition of pointed, movable shapes
§10 Approximative n-connectedness for movable pointed compacta
§11 n-movability
§12 A-movability 

CHAPTER VI EXTENSIOSN OF W-SEQUENCES AND S-SEQUENCES 

§1 Extensions and restrictions
§2 W-retractions and S-retractions (rel. M)
§3 W-retractions and W-retracts
§4 Deformation W-retracts
§5 Homomorphisms induced by W-retractions
§6 FAR(M)-spaces and FANR(M)-spaces
§7 Components of FANR(M)-spaces
§8 Cartesian product of FAR(M)-spaces
§9 Cartesian product of FANR(M)-spaces
§10 Suspension of FAR(M)-spaces and of FANR(M)-spaces
§11 On the union of two FANR(M)-spaces
§12 On the union of two FAR(M)-spaces 

CHAPTER VII SHAPES OF COMPACTA 

§1 Basic concepts of the theory of shape for compacta
§2 A condition characterizing the homotopy of two fundamental sequences
§3 The chimney lemma
§4 Extension of homotopic fundamental sequences
§5 Shapes of components of a compactum
§6 Pairs (X,Xo) with X є AR
§7 Shapes of plane continua
§8 Shapes of plane compacta
§9 Majorants for systems of compact shapes
§10 Fundamental dimension
§11 Homologous fundamental sequences
§12 Shapes of quasi-homeomorphic compacta
§13 Quasi-homeomorphic compacta with different shapes
§14 Two Jemmas
§15 Shapes of pointed compacta
§16 Lemma of Hurewicz
§17 A lemma of approximatively q-connected pointed compacta
§18 The modified homomorphism of Hurewicz
§19 Modified theorem of Hurewicz 

CHAPTER VIII FAR-SPACES AND FANR-SPACES 

§1 Elementary properties of FAR-spaces and of FANR-spaces
§2 Shape invariance of FAR-spaces and of FANR-spaces
§3 A special kind of fundamental retractions
§4 Fundamental retracts of approximatively n-connected and of apprximatively contractivle compacta
§5 Some remarks on the extendability of fundamental sequences
§6 A condition characterizing FAR-sets
§7 Strong movability
§8 Decreasing sequences of ANR-sets
§9 Plane FANR-sets
§10 Finite dimensional FAR-sets 

CHAPTER IX ARN-SEQUENCES AND THE THEORY OF SHAPE 

§1 ANR-sequences and π-maps
§2 Homotopy domination and homotopy equivalence for ANR-sequences
§3 π-maps associated with maps
§4 Inclusion ANR-sequences
§5 ANR-sequences and teh shape of compacta
§6 Some condiotions characterizing fundamental dimension
§7 Direct sequences of abelian groups
§8 Cohomology groups
§9 Shape-invariance of the cohomology groups of compacta 

CHAPTER X ON COMPLEMENTS OF COMPACTA LYING IN Q OR IN Eâ?¿ 

§1 Preliminaries
§2 Two lemmas
§3 Embedding of compacta with the same shapes in E� or in Q
§4 Completely equivalent pointed spaces
§5 Shapes of compact subsets of E�
§6 Two isomorphic categories
§7 Characterizing shapes of compacta in terms of embedding in Q
§8 On the shape of some decomposition spaces 

CHAPTER XI POSITIONS OF SETS IN SPACES 

§1 Homotopy position
§2 Similarity of pairs of space
§3 Position of a set in a space
§4 Position of closed subsets
§5 Examples and problems
§6 Multiplication of a position by a W-shape
§7 Suspension of positions
§8 Similar decreasing sequences of sets
§9 Positions of continua in the plane 

CHAPTER XII A SURVEY OF VARIOUS RESULTS AND PROBLEMD OF SHAPE THEORY 

§1 Axiomatic approach to shape theory
§2 Theory of shape for non-compact spaces
§3 Movable spaces
§4 Fundamental groups
§5 Classification of shapes
§6 Order properties for shapes
§7 Fundamental dimension
§8 FANR-spaces
§9 Shape properties of decomposition spaces
§10 Embedding problems
§11 Decomposition of shapes into products
§12 Problem of reasonable representatives of shapes
§13 Problems of characterization of shapes
§14 Homotopically compact spaces 

 
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