Convergence foundations of topology:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
New Jersey
World Scientific
[2016]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xix, 548 Seiten Illustrationen, Diagramme |
| ISBN: | 9789814571517 9789814571524 |
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| 245 | 1 | 0 | |a Convergence foundations of topology |c Szymon Dolecki (Mathematical Institute of Burgundy, France), Frédéric Mynard (New Jersey City University, USA) |
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Record in the Search Index
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|---|---|
| adam_text | Titel: Convergence foundations of topology
Autor: Dolecki, Szymon
Jahr: 2016
Contents
Preface vii
I. Introduction 1
1. Preliminaries and conventions................................4
2. Premetrics and balls ..........................................6
3. Sequences ......................................................9
4. Cofiniteness....................................................13
5. Quences ........................................................14
6. Almost inclusion.....................................17
7. When premetrics and sequences do not suffice..............19
7.1. Pointwise convergence......................19
7.2. Riemann integrals........................................23
II. Families of sets 27
1. Isotone families of sets........................................27
2. Filters ..........................................................29
2.1. Order......................................................30
2.2. Free and principal filters..................................33
2.3. Sequential filters..........................................35
2.4. Images, preimages, products ............................37
3. Grills............................................................39
4. Duality between filters and grills..............................41
5. Triad: filters, filter-grills and ideals..........................42
6. Ultrafilters......................................................43
7. Cardinality of the set of ultrafilters..........................46
8. Remarks on sequential filters..................................48
8.1. Countably based and Frechet filters....................48
8.2. Inhma and products of filters............................51
9. Contours and extensions......................................53
III. Convergences 55
1. Definitions and first examples................................55
2. Preconvergences on finite sets................................60
2.1. Preconvergences on two-point sets......................60
2.2. Preconvergences on three-point sets....................62
3. Induced (pre) convergence......................................64
4. Premetrizable convergences....................................65
5. Adherence and cover ..........................................67
6. Lattice of convergences........................................70
7. Finitely deep modification....................................71
8. Pointwise properties of convergence spaces..................72
9. Convergences on a complete lattice..........................75
IV. Continuity 79
1. Continuous maps..............................................79
2. Initial and final convergences..................................82
3. Initial and final convergences for multiple maps............86
4. Product convergence..........................................89
4.1. Finite product............................................89
4.2. Infinite product ..........................................91
5. Functional convergences ......................................92
6. Diagonal and product maps..................................94
6.1. Diagonal map ............................................94
6.2. Product map..............................................95
7. Initial and final convergences for product maps ............96
8. Quotient........................................................97
9. Convergence invariants........................................102
9.1. Premetrizability, metrizability ..........................103
9.2. Isolated points, paving number, finite depth............104
9.3. Characters and weight..................................105
9.4. Density and separability................................109
V. Pretopologies 115
1. Definition and basic properties..........................115
2. Principal adherences and inherences........................121
3. Open and closed sets, closures, interiors, neighborhoods . . 128
4. Topologies...............135
4.1. Topological modification.................139
4.2. Induced topology..........................................443
4.3. Product topology ........................................444
5. Open maps and closed maps..................................147
6. Topological defect and sequential order......................149
6.1. Iterated adherence and topological defect..............149
6.2. Sequentially based convergence and sequential order . 153
VI. Diagonality and regularity 161
1. More on contours..............................................161
2. Diagonality......................................................163
2.1. Various types of diagonality..............................165
2.2. Diagonal modification....................................170
3. Self-regularity..................................................171
4. Topological regularity..........................................176
5. Regularity with respect to another convergence ............177
VII. Types of separation 179
1. Convergence separation........................................179
2. Regularity with respect to a family of sets..................182
3. Functionally induced convergences............................184
4. Real-valued functions..........................................187
5. Functionally closed and open sets............................188
6. Functional regularity (aka complete regularity)..............191
7. Normality ......................................................197
8. Continuous extension of maps................................205
9. Tietze s extension theorem....................................209
VIII. Pseudotopologies 213
1. Adherence, inherence..........................................213
2. Pseudotopologies ..............................................216
3. Pseudotopologizer..............................................218
4. Regularity and topologicity among pseudotopologies .... 221
5. Initial density in pseudotopologies............................223
6. Natural convergence............................................225
7. Convergences on hyperspaces..................................226
IX. Compactness 231
1. Compact sets .........................
2. Regularity and topologicity in compact spaces..............238
3. Local compactness..............................................240
4. Topologicity of hyperspace convergences ....................245
5. The Stone topology............................................247
6. Almost disjoint families........................................252
7. Compact families..............................................256
8. Conditional compactness......................................261
8.1. Paratopologies............................................262
8.2. Countable compactness..................................262
8.3. Sequential compactness..................................265
9. Upper Kuratowski topology..................................271
10. More on covers ................................................272
11. Cover-compactness............................................275
12. Pseudocompactness............................................279
X. Completeness in metric spaces 283
1. Complete metric spaces........................................283
2. Completely metrizable spaces................................288
3. Metric spaces of continuous functions........................290
4. Uniform continuity, extensions, and completion ............292
XI. Completeness 297
1. Completeness with respect to a collection....................297
2. Cocompleteness................................................299
3. Completeness number..........................................302
4. Finitely complete convergences................................305
5. Countably complete convergences............................306
6. Preservation of completeness..................................307
7. Completeness of subspaces....................................309
8. Completeness of products....................................311
9. Conditionally complete convergences......................314
10. Baire property....................................315
11. Strict completeness............................317
XII. Connectedness 3ig
1. Connected spaces..............................................319
2. Path connected and arc connected spaces....................326
3. Components and quasi-components..........................328
4. Remarks on zero-dimensional spaces..........................333
XIII. Compactifications 335
1. Introduction....................................................335
2. Compactifications of functionally regular topologies .... 338
3. Filters in lattices ..............................................343
4. Filters in lattices of closed and functionally closed sets . . 345
5. Maximality conditions ........................................347
6. Cecli-Stone compactification..................................349
XIV. Classification of spaces 355
1. Modifiers, projectors, and coprojectors......................355
2. Functors, reflectors and coreflectors..........................360
3. Adherence-determined convergences..........................363
3.1. Reflective classes..........................................364
3.2. Composable classes of filters ............................366
3.3. Conditional compactness................................368
4. Convergences based in a class of filters......................370
5. Other Fo-composable classes of filters........................373
6. Functorial inequalities and classification of spaces..........375
7. Reflective and coreflective hulls ..............................380
8. Conditional compactness and cover-compactness............385
XV. Classification of maps 389
1. Various types of quotient maps ..............................389
1.1. Remarks on the quotient convergence..................389
1.2. Topologically quotient maps ............................390
1.3. Hereditarily quotient maps..............................393
1.4. Quotient maps relative to a reflector....................395
1.5. Biquotient maps..........................................396
1.6. Almost open maps........................................397
1.7. Countably biquotient map ..............................398
2. Interactions between maps and spaces........................398
3. Compact relations..............................................400
4. Product of spaces and of maps................................404
XVI. Spaces of maps 411
1. Evaluation and adjoint maps..................................412
2. Adjoint maps on spaces of continuous maps ........415
3. Fundamental convergences on spaces of continuous maps . 416
4. Point wise convergence..........................................417
5. Natural convergence............................................420
5.1. Continuity of limits......................................421
5.2. Exponential law.....................423
5.3. Finer subspaces and natural convergence.......425
5.4. Continuity of adjoint maps..............................427
5.5. Initial structures for adjoint maps......................429
6. Compact subsets of function spaces (Ascoli-Arzela) .... 431
XVII. Duality 437
1. Natural duality........................437
2. Modified duality................................................443
3. Concrete characterizations of bidual reflectors..............449
4. Epitopologies ..................................................450
5. Functionally embedded convergences........................451
6. Exponential hulls and exponential objects .........453
7. Duality and product theorems................................459
8. Non-Frechet product of two Frechet compact topologies . . 466
9. Spaces of real-valued continuous functions..................469
9.1. Cauchy completeness ....................................469
9.2. Completeness number....................................470
9.3. Character and weight....................................471
XVIII. Functional partitions and metrization 475
1. Introduction..............................................475
2. Perfect normality....................................475
3. Pseudometrics..................................478
4. Functional covers and partitions............................481
5. Paracompactness..............................488
6. Fragmentations of partitions of unity......................491
7. Metrization theorems................494
A. Set theory 497
1. Axiomatic set theory..........................................497
2. Basic set theory................................................499
3. Natural numbers ..............................................501
4. Cardinality......................................................502
5. Continuum......................................................507
6. Order............................. ¦ 509
7. Lattice..........................................................510
8. Well ordered sets ..............................................512
9. Ordinal numbers................................................514
10. Ordinal arithmetic ............................................517
11. Ordinal-cardinal numbers......................................519
Bibliography 523
List of symbols 529
Index 541
|
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| spelling | Dolecki, Szymon Verfasser (DE-588)1112147144 aut Convergence foundations of topology Szymon Dolecki (Mathematical Institute of Burgundy, France), Frédéric Mynard (New Jersey City University, USA) New Jersey World Scientific [2016] xix, 548 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Topology Textbooks Convergence Topological groups Konvergenz (DE-588)4032326-2 gnd rswk-swf Topologie (DE-588)4060425-1 gnd rswk-swf Topologie (DE-588)4060425-1 s Konvergenz (DE-588)4032326-2 s DE-604 Mynard, Frédéric 1973- Verfasser (DE-588)1112147268 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029266553&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dolecki, Szymon Mynard, Frédéric 1973- Convergence foundations of topology Topology Textbooks Convergence Topological groups Konvergenz (DE-588)4032326-2 gnd Topologie (DE-588)4060425-1 gnd |
| subject_GND | (DE-588)4032326-2 (DE-588)4060425-1 |
| title | Convergence foundations of topology |
| title_auth | Convergence foundations of topology |
| title_exact_search | Convergence foundations of topology |
| title_full | Convergence foundations of topology Szymon Dolecki (Mathematical Institute of Burgundy, France), Frédéric Mynard (New Jersey City University, USA) |
| title_fullStr | Convergence foundations of topology Szymon Dolecki (Mathematical Institute of Burgundy, France), Frédéric Mynard (New Jersey City University, USA) |
| title_full_unstemmed | Convergence foundations of topology Szymon Dolecki (Mathematical Institute of Burgundy, France), Frédéric Mynard (New Jersey City University, USA) |
| title_short | Convergence foundations of topology |
| title_sort | convergence foundations of topology |
| topic | Topology Textbooks Convergence Topological groups Konvergenz (DE-588)4032326-2 gnd Topologie (DE-588)4060425-1 gnd |
| topic_facet | Topology Textbooks Convergence Topological groups Konvergenz Topologie |
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