Stavrakas / Allen Studies in Topology

1. Auflage 2014
ISBN: 978-1-4832-5911-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 672 Seiten, Web PDF

Studies in Topology is a compendium of papers dealing with a broad portion of the topological spectrum, such as in shape theory and in infinite dimensional topology. One paper discusses an approach to proper shape theory modeled on the 'ANR-systems' of Mardesic-Segal, on the 'mutations' of Fox, or on the 'shapings' of Mardesic. Some papers discuss homotopy and cohomology groups in shape theory, the structure of superspace, on o-semimetrizable spaces, as well as connected sets that have one or more disconnection properties. One paper examines 'weak' compactness, considered as either a strengthening of absolute closure or a weakening of relative compactness (subject to entire topological spaces or to subspaces of larger spaces). To construct spaces that have only weak properties, the investigator can use the various productivity theorems of Scarborough and Stone, Saks and Stephenson, Frolik, Booth, and Hechler. Another paper analyzes the relationship between 'normal Moore space conjecture' and productivity of normality in Moore spaces. The compendium is suitable for mathematicians, physicists, engineers, and other professionals involved in topology, set theory, linear spaces, or cartography.

Autoren/Hrsg.

Stavrakas, Nick M.

Allen, Keith R.

Inhaltsverzeichnis

1;Front Cover;1
2;Studies in Topology;4
3;Copyright Page;5
4;Table of Contents;8
5;Contributors;12
6;Preface;16
7;Acknowledgments;18
8;Birth of the Polish School of Mathematics;20
8.1;References;23
9;Chapter 1. Alternative Approaches to Proper Shape Theory;24
9.1;References;49
10;Chapter 2. On the Existence and Uniqueness Theorems of R. S. Pierce for Extensions of Zero-Dimensional Compact Metric Spaces;52
10.1;References;64
11;Chapter 3. Mapping Continua Onto the Cone Over the Cantor Set;66
11.1;References;67
12;Chapter 4. Nearness Spaces and Extensions of Topological Spaces;70
12.1;0. Preliminaries;70
12.2;1. Topological extensions;72
12.3;2. Functorial relationships between Near and Ex;74
12.4;3. Bunch determined nearness spaces;80
12.5;4. Remarks and questions;86
12.6;References;88
13;Chapter 5. On Several Problems of the Theory of Shape;90
13.1;1. To find a pure definition of e(X);93
13.2;2. Is it true that Sh(X) = (Y) implies that e(X) = e(Y)?;93
13.3;References;101
14;Chapter 6. Some Results on (E,ßE)-Compactness;104
14.1;References;115
15;Chapter 7. Toroidal Decompositions of Manifolds Yield Factors of Manifolds;116
15.1;INTRODUCTION;116
15.2;2. TERMINOLOGY;117
15.3;3. PRELIMINARY RESULTS;118
15.4;4. MAIN LEMMA;121
15.5;References;132
16;Chapter 8. Homotopy and Cohomoiogy Groups in Shape Theory;134
16.1;References;142
17;Chapter 9. The Structure of Superspace;144
17.1;I. The Configuration Space of a Physical System;144
17.2;II. Cosmology;146
17.3;III. Superspaoe;148
17.4;References;156
18;Chapter 10. Some Notes on Multifunctions;158
18.1;References;160
19;Chapter 11. Connected Sets With a Finite Disconnection Property;162
19.1;1. INTRODUCTION;162
19.2;2. DEFINITIONS AND TERMINOLOGY;163
19.3;3. BASIC PROPERTIES OF .-SPACES;164
19.4;4. LOCALLY CONNECTED .-SPACES;170
19.5;5. COMPACT .-SPACES;183
19.6;6. DECOMPOSITION OF .-SPACES;191
19.7;References;195
20;Chapter 12. Applications of Collectionwise Hausdorff;198
20.1;References;199
21;Chapter 13. On o-semimetrizable Spaces;202
21.1;1) Introduction;202
21.2;2) Characterizations of o-semimetrizable spaces;203
21.3;References;210
22;Chapter 14. Characterizing Topological Properties;212
22.1;1. Introduction;212
22.2;2. Terminology;212
22.3;3. Main Results;213
22.4;References;219
23;Chapter 15. . Connectivity in the Plane;220
23.1;References;224
24;Chapter 16. On Continuous Extenders;226
24.1;References;234
25;Chapter 17. On a Notion of Weak Compactness in Non-Regular Spaces;238
25.1;1. Introduction;238
25.2;2. Characterizations and Properties;241
25.3;3. Examples;246
25.4;4. Compactification;248
25.5;5. Generalizations of Sequential Compactness;249
25.6;6. Motivation;257
25.7;7. Open Problems;258
25.8;References;259
26;Chapter 18. Actions of Locally Compact Groups With Zero on Manifolds;262
26.1;References;276
27;Chapter 19. Non-Continuous Retracts;278
27.1;References;283
28;Chapter 20. The Nielsen Numbers and Fiberings;286
28.1;0. Introduction;286
28.2;1. Principal Tk -bundle over lens spaces;288
28.3;2. The Nielsen and Lefschetz numbers;291
28.4;3. The fixed point property for a fiber preserving mapping;295
28.5;References;297
29;Chapter 21. Maps of ANR's Determined on Null Sequences of AR's;300
29.1;References;307
30;Chapter 22. Two Vietoris-type isomorphism theorems in Borsuk's theory of shape, concerning the Vietoris-Cech homology and Borsuk's fundamental groups;308
30.1;Introduction;308
30.2;0. Preliminaries;309
30.3;1. The isomorphism theorem for Borsuk's fundamental;310
30.4;2. The case of the pointed sequence generated by a map;317
30.5;3. A further application;320
30.6;4. The isomorphism theorem for Vietoris homology groups;322
30.7;References;336
31;Chapter 23. Uniformly Pathwise Connected Continua;338
31.1;INTRODUCTION;338
31.2;1. PRELIMINARIES;340
31.3;2. IMPROVING UNIFORM FAMILIES OF PATHS BY REPARAMETERIZATION;341
31.4;3. UNIFORMLY PATHWISE CONNECTED CONTINUA;344
31.5;References;347
32;Chapter 24. Several Problems of Continua Theory;348
32.1;1. CONFLUENCY AND RATIONAL CONTINUA;348
32.2;2. CONFLUENCY AND CLASS A;349
32.3;3. CONFLUENCY RELATIVE TO LOCALLY CONNECTED CONTINUA;350
32.4;References;351
33;Chapter 25. A Characterization of Local Connectivity in Dendroids;354
33.1;1. Introduction;354
33.2;2. Characterizations of dendrites;355
33.3;References;361
34;Chapter 26. A Survey of Embedding Theorems for Semigroups of Continuous Functions;362
34.1;1 . INTRODUCTION;362
34.2;2. THE CASE WHERE Y IS DISCRETE;362
34.3;3. THE CASE WHERE Y IS NOT DISCRETE;366
34.4;4. CONCLUSION;373
34.5;References;376
35;Chapter 27. THE HUREWICZ AND WHITEHEAD THEOREMS IN SHAPE THEORY;378
35.1;1. Pro-categories;379
35.2;2. Pro-groups;380
35.3;3. Pro-homotopy category of CW-complexes;381
35.4;4. The shape category;382
35.5;5. The Hurewicz theorem;383
35.6;6. The Whitehead theorem;384
35.7;7. Homology versions of the Whitehead theorem;386
35.8;References;387
36;Chapter 28. One-Dimensional Shape Properties and Three-Mamfolds;390
36.1;1. INTRODUCTION;390
36.2;2. IMAGES OF 1-MOVABLE CONTINUA;392
36.3;3. DESCRIBING COMPACTA IN THREE-MANIFOLDS;398
36.4;References;403
37;Chapter 29. Discontinuous Gd Graphs;406
37.1;Introduction;406
37.2;1. Sequences of open sets characterizing functions of Baire Class 1;408
37.3;2. Examples;413
37.4;References;415
38;Chapter 30. Some Basic Connectivity Properties of Whitney Map Inverses in C(X);416
38.1;1. Introduction and basic definitions;416
38.2;2. A result of Krasinkiewiaz;420
38.3;3. Main results;421
38.4;4. Problems;430
38.5;References;432
39;Chapter 31. One-Point Compactifications of Q-manifold Factors and Infinite Mapping Cylinders;434
39.1;I. Introduction;434
39.2;II. Basic Results;436
39.3;References;448
40;Chapter 32. Some Surprising Base Properties in Topology;450
40.1;INTRODUCTION;450
40.2;1. Non-archimedean spaces;450
40.3;2. Productively non-archimedean and wµ -metrizable spaces;455
40.4;3. Proto-metrizdble spaces;458
40.5;4. Proto-uniformities and proto-metrics;462
40.6;5. Basically screenable spaces;465
40.7;6. Countable-dimensional spaces;467
40.8;7. Applications to dimension theory;469
40.9;References;471
41;Chapter 33. Some Topological Questions Related to Open Mapping and Closed Graph Theorems;474
41.1;References;478
41.2;Completeness in Aronszajn Spaces;480
41.3;1. INTRODUCTION;480
41.4;2. SETS OF INTERIOR CONDENSATION;483
41.5;3. ARONSZAJN SPACES AND MOORE SPACES;485
41.6;References;487
42;Chapter 34. Projectives in the Category of Ordered Spaces;490
42.1;1. INTRODUCTION;490
42.2;2. DEFINITIONS AND NOTATIONS;491
42.3;3. MAIN RESULTS;493
42.4;References;500
43;Chapter 35. On the Productivity of Normality in Moore Spaces;502
43.1;I. Review of product results;502
43.2;II. Recent metrization results;503
43.3;References;506
44;Chapter 36. A Metrization Theorem for Normal Moore Spaces;508
44.1;References;511
45;Chapter 37. Expansions of Topologies by Locally Finite Collections;512
45.1;1. Introduction;512
45.2;2. Preliminaries;513
45.3;3. Main Results;514
45.4;References;516
46;Chapter 38. Some Approximation Theorems for Inverse Limits;518
46.1;1. introduction;518
46.2;2. The approximation theorem;520
46.3;References;528
47;Chapter 39. The Metrizability of Normal Moore Spaces;530
47.1;References;537
48;Chapter 40. Toward a Product Theory for Orthocompactness;540
48.1;0. Notation and Terminology;540
48.2;1. Products with a compact factor;543
48.3;2. Products with a metric factor;545
48.4;3. Products of Ordinals;549
48.5;4. Examples;555
48.6;5. Addendum;558
48.7;References;559
49;Chapter 41. Movable Continua and Shape Retracts;562
49.1;References;566
50;Chapter 42. n-adic Decompositions and Retracts;568
50.1;1. Introduction and terminology;568
50.2;2. A survey;568
50.3;3. n-adic Antoine's necklaces and n-adic wreaths;569
50.4;References;573
51;Chapter 43. Embedding Characterizations for Collectionwise Normality and Expandability;576
51.1;Section 1. Introduction;576
51.2;Section 2. An Embedding Characterization for Collectionwise Normal Spaces;577
51.3;Section 3. Characterizations for Expandable Spaces;578
51.4;References;580
52;Chapter 44. Extending Maps from Products;582
52.1;References;587
53;Chapter 45. Dense Subsemigroups of Semigroups Of Continuous Selfmaps;588
53.1;INTRODUCTION;588
53.2;References;594
54;Chapter 46. Banach Spaces With Banach-Stone Property;596
54.1;Section 1. Basic Concepts;597
54.2;Section 2. Examples;597
54.3;References;603
55;Chapter 47. PRIMITIVE STRUCTURES IN GENERAL TOPOLOGY;604
55.1;I. INTRODUCTION;604
55.2;2. PRIMITIVE SEQUENCES;605
55.3;3. SOME KNOWN SPACES IN TERMS OF PRIMITIVE SEQUENCES;610
55.4;4. A GENERAL APPROACH TO DEFINING TOPOLOGICAL STRUCTURE VIA PRIMITIVE SEQUENCES;613
55.5;5. PRIMITIVE STRUCTURE OF (COUNTABLY) COMPACT TYPE; RELATIONS TO OTHER SPACES;615
55.6;References;620
56;Chapter 48. Recent Developments in Dendritic Spaces and Related Topics;624
56.1;1. Introduction;624
56.2;2. A catalog of dendritic properties According to the definitions in the preceeding section;627
56.3;3. Comb spaces; completion of the proof of Theorem 9;639
56.4;4. Further partial order characterizations;641
56.5;5. Locally connected spaces;644
56.6;7. Trees;662
56.7;8. Fixed point theorems for dendritic spaces;663
56.8;9. Concluding remarks;667
56.9;References;668
57;Index;672