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Mill J.V. — The Infinite-Dimensional Topology of Function Spaces
Mill J.V. — The Infinite-Dimensional Topology of Function Spaces

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Название: The Infinite-Dimensional Topology of Function Spaces

Автор: Mill J.V.

Аннотация:

In this book we study function spaces of low Borel complexity. Techniques from general topology, infinite-dimensional topology, functional analysis and descriptive set theory are primarily used for the study of these spaces. The mix of methods from several disciplines makes the subject particularly interesting. Among other things, a complete and self-contained proof of the Dobrowolski-Marciszewski-Mogilski Theorem that all function spaces of low Borel complexity are topologically homeomorphic, is presented.


In order to understand what is going on, a solid background in infinite-dimensional topology is needed. And for that a fair amount of knowledge of dimension theory as well as ANR theory is needed. The necessary material was partially covered in our previous book `Infinite-dimensional topology, prerequisites and introduction'. A selection of what was done there can be found here as well, but completely revised and at many places expanded with recent results. A `scenic' route has been chosen towards the Dobrowolski-Marciszewski-Mogilski Theorem, linking the results needed for its proof to interesting recent research developments in dimension theory and infinite-dimensional topology.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 2001

Количество страниц: 644

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$C*_p(X)$      368 371 394 396—399 418 421 425 426 445 590 591
$C_0$      8 354 360 435 436
$C_P(X)$      4 368—375 377—379 383—387 393—399 402—404 406 407 409—415 425—428 441 442 444—447 452 455 456 590—592
$F_{\sigma \delta}$      377 442 445 446 518 592
$F_{\sigma \delta}$, absolute      344 518—520
$f_{\sigma}$      87 227 230 371 466 517 518 522 595
$F_{\sigma}$, absolute      518
$G_{\delta \sigma}$      518
$G_{\delta \sigma}$, absolute      344 518—520
$G_{\delta}$      35 172 175 217 220 221 229 231 238 257 373 375 127 428 434 454 464 466 467 480 483 485 493 494 517 526 590
$G_{\delta}$, absolute      518
$G_{\delta}$, enlargement      175 304
$G_{\delta}$, selection      217
$l*$-equivalent      370 418 419 421 425 456
$L_{\varepsilon}$-enlargement      189
$\delta$-multiplicative      517 522
$\mathcal{A}$-operation      594
$\mathcal{F}$-ultrafilter      495 496
$\mathcal{F}_{\sigma \delta}$-absorber      350 353 360 361 442 444
$\mathcal{F}_{\sigma}$-absorber      347 350 361
$\mathfrak{U}$-close      263
$\mathfrak{U}$-map      135 136 166 169 176
$\sigma Z$-set      307 334 361 436 441
$\Sigma$      257
$\sigma$-addilive      517 522
$\sigma$-algebra      455 458 531
$\sigma$-compact      58 66 226 228 339 347 361 411 412 426 479 518 519 581
$\sigma$-compact, nowhere      581
$\sigma$-discrete      369
$\varepsilon$-map      34 66 67 90 135
Aarts      584 587 594 597
Absolute (Neighborhood) Extensor      25
Absolute neighborhood retract      24—29 38 39 124 145 148 263—265 270 274 276 284 288 289 301—305 580 581 588
Absolute Neighborhood Retract, pair      266 267 273 276 277 285 290 301 304 305 588
Absolute retract      24—27 29 39 145 276 288 289 294 301 305 580
Absolute Retract, hyperspace      292 588
Absolute Retract, pair      266 288 290
Absolute Retract, pair, hyperspace      292
Absolute value      512
Absorber      347 348 589
Absorber Bestvina — Mogilski type      434 589 590
Absorber Dijkstra — van Mill — Mogilski type      347
Absorbing sequence      347 348
Absorbing system      346 347
Accumulation point      462
Addition theorem      178
Adjan      587 597
Adjunction space      507 508
Affine combination      111 114
Affine coordinates      114 125 126 134
Affine function      112—114
Affine function, continuous      113
Affine hull      111
Affine subspace      111—113 124
Alexandroff      253 254 582 584 587 593 597
Alexandroff compactification      472
Alexandroff problem      253 254 587
Almost zero-dimensional      167 168 189 192 583
Analytic set      see set analytic
Anderson      9 325 579 580 584 585 588—590 597 598
Anderson Theorem      9 579 588
Antipodal map      148 238 461
Antipodal points      149 461
Antipodal, preserving      149
Approximation Theorem Brown      90
Approximation Theorem Freudenthal      137
Arens      396 579 598
Arhangel'skii      367 372 411 426 451 590 591 598 603
Arveson      455 598
Aull      579 598
Baars      x 367 411 425 434 591 592
Baire category theorem      482
Baire space      3 36 66 78 79 373 378 379 383—385 387 391 392 446 482—485 526 580 591—593
Baire space, hereditary      393 446
Baire space, not topologically complete      4&4
Ball, closed      459
Ball, open      459
Banach      579 598
Banach space      3 4 9 10 12 17 21 370 394 579
Banakh      426 437 445 588 590 598
Bandt      594 598
Barit      589 598
Barov      x
Bartle      579 598
Barycenter      125 126 128 129 138 139
Barycentric subdivision      129—132
Barycentric triangulation      129 130
Base      41 64 122 153 167 177 238 239 285 288 369 465 467 497 507
Base, closed sets      494 497
Base, collection of sets      441
Base, countable      593
Base, Ellentuck topology      250 251
Base, hyperspace      101
Base, inverse limit      81
Base, local      19 183 368 467 468
Base, Wallman      93 185 494—500
Basic core set      342 343
Becker      49 580 598
Bendixson      592
Bennett      580 598
Berney      593 594 599
Bernstein set      580
Bessaga      371 588 589 599
Bestvina      434 588—590 599
bing      215 580 585 589 593 597 599
Bing compactum      210 212—215
Bing Shrinking Criterion      67 580
Blaszczyk      586 599
Blumenthal      579 599
Bonding map      80 84 93
Borel      594
Borel complexity      445 590
Borel set      80 373 455 456 517—519 523 526 586 590 595
Borel set, absolute      18 437 456 526 592
Borel set, homogeneous      581
Borel set, rigid      581
Borsuk      301 580 581 583 585—588 599
Borsuk Antipodal Theorem      149
Borsuk Example      301 .380
Borsuk Homotopy Extension Theorem      38
Borsuk Homotopy Extension Theorem, controlled      265
Borsuk Problem      588
Borsuk — Ulam theorem      149 174
Boundary      20 125 196 460 464
Boundary, geometric      114
Boundary, preserving homeomorphism      311
Boundary, set      589
Bounded component      505
Bounded function      29 174 368 445
Bounded metric      459 463
Bounded set in a normed linear space      3 7 20 21
Bounded subspace      407 410 417 478
Bourbaki      585 593 599
Bowers      586 588 599 600
Brouwer      221—223 580 582 600
Brouwer $\mathbb{R}^n \neq \mathbb{R}^m$      166
Brouwer Dimensionsgrad      221
Brouwer fixed-point theorem      143 145 148 149 260 587 594
Brouwer Invariance of Domain Theorem      197
Brown      579 600
Burckol      594 600
C(X)      4 5 19 20 31 368 370—372 393 394 418
C-imbedded      410
Calbrix      446 590 592 600
Cantor      580 592 600
Cantor set      42—48 75 176 229 238 254 257 387 388 434 436 581
Cantor — Bendixson Theorem      467
Cantor-manifold      207 208
Capset      329—331 333 334 337 338 341—343 347 354 589
Carrier      119 134 284
Cauchy sequence      4 33 58 96 464 468 469
Cauty      24 26 264 367 414 426 445 446 456 580 583 590 592 598 600
Cauty Examples      see Example(s) Cauty
Cech      583 584 600
Cech — Stone compactification      368 590
Centered      495
Centered, maximal      495
Chapman      588 589 597 600
Characterization, $C_P(X)$ Baire      379
Characterization, $C_p(X)$ metrizable      369
Characterization, $LC^n$      296 298
Characterization, $LC^n$ and $C^n$      300
Characterization, $l^2$-manifold      588
Characterization, $S_p(X) \approx B(Q)^{\infty}$      444
Characterization, $\mathbb{I}$      51
Characterization, $\mathbb{P}$      76 77
Characterization, $\mathbb{Q}$      76
Characterization, A(N)R-pair      267 277
Characterization, absolute $F_{\sigma \delta}$      519
Characterization, analytic set      77
Characterization, ANR      284
Characterization, AR      39 289
Characterization, boundary point      196
Characterization, Cantor set      43
Characterization, Capset      334 337 590
Characterization, Compact      46O
Characterization, dimension      160 166 174 182 195
Characterization, finite dimensional manifold      588
Characterization, hereditarily      94
Characterization, homogeneous Borel set      77
Characterization, indecomposable continuum      86
Characterization, inessential map      514
Characterization, meager filter      389 434
Characterization, Menger manifold      588
Characterization, Q      294
Characterization, Q-manifold      588 589
Characterization, topologically complete      450
Characterization, zero-dimensional      58 581
Chatyrko      587 600
Chigogidze      588 600 603
Choquct      384 601
Christensen      426 601
Class of spaces, closed hereditary      344
Class of spaces, topological      344
Clopen      41 50 57 58 63 443 460 500
Closed ball      459
Closed base      494
Closed map      see map closed
Closed shrinking      485
Closure      460
Coanalytic set      594
Cohen      384 601
COLOR      187
Color number      250
Color, open      251
Color, open or closed      251
Colorable      188
Coloring      187 188 238 241 242 244 245 248 249 251 586
Combinatorially equivalent      123
Comfort      590 592 601
Compact-open topology      19 20 31
Compactification      57 64 87 93 174 182—184 188 193 207 234 253 257 472 473 478 500 510 516
Compactification, Alexandroff      472
Compactification, Cech — Stone      368 590
Compactification, dimension preserving      183
Compactification, equivalent      472
Compactification, one-point      472
Compactification, remainder      183
Compactification, Wallman      93 185 494 498 499 593
Compactness, deficiency      183 204
Compactness, degree      182 183
Compactum Bing      see Bing compactum
Compactum Henderson      see Henderson compactum
Complete with respect to a norm      16
Complete, metric      see metric complete
Complete, topologically      see topologically complete
Completely Ramsey set      239
Complex conjugate      512
Component      37 156 167 204 209 285 467 500 501 505 507
Component path      503
Composant      87 94
Cone      21 39 70 89 302 515 516 581
Connected      54 55 57 58 64 85 102 156 206 207 209 473 503—505
Connected in dimension n      295
Connected, locally      see locally connected
Connected, path      see path-connected
Continuous image, $\mathbb{P}$      77
Continuous image, C      46
Continuous image, complete space      522
Continuous image, Menger curve      585
Continuous image, one-dimensional space      203
Continuous image, zero-dimensional space      57 58
Continuous logarithm      513
Continuum      37 51 58 85 94 95 105 176 209 210 216 261 501 502 505 515 581 593
Continuum Hypothesis      47 384 393
Continuum, $sin(1/x)$      29 57 503 505 517
Continuum, cook      447
Continuum, decomposable      86
Continuum, from A to B      206 504
Continuum, hereditarily indecomposable      87 94 106 108 212 213 258 260 587
Continuum, hereditarily indecomposable, homogeneous      215 216
Continuum, indecomposable      86 87
Continuum, indecomposable, homogeneous      215
Continuum, Peano      58 226 292 294 295 300 340 585 588
Continuum, unicoherent      226 514
Continuum-connected      206 207 209
Contractible      29 39 70 146—148 274 285 289 301 511 512 515 516 580
Contractible, locally      see locally contractible
Contraction      511
Convex      1 2 15 16 19 20 23 25 29 124 125 148 288 581 582
Convex, combination      1 2
Convex, hull      1
Convex, metric      477 479 588
Cook      447 601
Coordinate, functions of a simplex      114
Coordinate, space of inverse sequence      80
Countable closed sum theorem      163
Countable dense homogeneous      63—66 580
Countable dense homogeneous, not homogeneous      63
Countable, dimensional      155 156 221 252 253 428
Countably, compact      473
Countably, continuous      257
Cover      458
Cover, locally finite      486
Cover, refinement      458
Cover, star-finite      135 136
Cover, star-refinement      460
Covering dimension      152 160 583
Criterion, Bing Shrinking      67
Criterion, Inductive Convergence      59 65 325
Curtis      291 325 580 584 588—590 598 601
Curtis — Schori — West Hyperspace Theorem      291 295
Cut      222 504
Cut point      51 504 505
Cut, need not be a partition      221 505
Daverman      588 594 601
de Bruijn      586 600
de Groot, ,J.      373 580 581 584 603 604
de Groot, J.A.M.      367 411 591 592 598
de Vries      x 582 612
Decomposable continuum      86
Deformation      510
Deformation property      334 337 343
Deformation through a subset      266—268 276 294 327 355 360 361
Degree compactness      182 183
Degree partition      153
Dellacherie      610
Derivative      417
Derived set      417
Descriptive complexity      373—375 377 446 590
1 2 3 4
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