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 Infima 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 ...................................... 143
4.3 Product topology ...................................... 144
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 ............................................... 231
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 ............................................ 319
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 Maximali ty conditions .................................... 347
6 Cech-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  0-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 Pointwise 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-Arzelд) ......... 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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