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Aarts J.M., Nishiura T. — Dimension and Extensions (North-Holland Mathematical Library)
Aarts J.M., Nishiura T. — Dimension and Extensions (North-Holland Mathematical Library)

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Название: Dimension and Extensions (North-Holland Mathematical Library)

Авторы: Aarts J.M., Nishiura T.

Аннотация:

Two types of seemingly unrelated extension problems are discussed in this book. Their common focus is a long-standing problem of Johannes de Groot, the main conjecture of which was recently resolved. As is true of many important conjectures, a wide range of mathematical investigations had developed, which have been grouped into the two extension problems. The first concerns the extending of spaces, the second concerns extending the theory of dimension by replacing the empty space with other spaces. The problems of De Groot concerned compactifications of spaces by means of an adjunction of a set of minimal dimension. This minimal dimension was called the compactness deficiency of a space. Early success in 1942 led De Groot to invent a generalization of the dimension function, called the compactness degree of a space, with the hope that this function would internally characterize the compactness deficiency which is a topological invariant of a space that is externally defined by means of compact extensions of a space. From this, the two extension problems were spawned. With the classical dimension theory as a model, the inductive, covering and basic aspects of the dimension functions are investigated in this volume, resulting in extensions of the sum, subspace and decomposition theorems and theorems about mappings into spheres. Presented are examples, counterexamples, open problems and solutions of the original and modified compactification problems.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1993

Количество страниц: 331

Добавлена в каталог: 10.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(\leq n)$-Inductionally embedded      282
$F_{\sigma}$-set      6
$G_{\delta}$-set      29
$\Delta$      (see “Roy’s example”)
$\delta$-cover      303
$\delta$-extension      302 6
$\delta$-function      301
$\delta$-homeomorphism      301
$\epsilon$-mapping      160
$\kappa(\cdot)$      288
$\mathcal{C}$-border cover      49
$\mathcal{C}$-kernel      49
$\mathcal{C}$-surplus      58
$\mathcal{P}$-border cover      94
$\mathcal{P}$-hull      109
$\mathcal{P}$-kernel      85
$\mathcal{P}$-stable value      100
$\mathcal{P}$-Surplus      85
$\mathcal{P}$-unstable      100
$\sigma$-locally finitely closed-additive in $\mathcal{U}$      142
$\sigma$-locally paracompact      158
Absolute $G'<s$-space      31
Absolute additive Borel class      115
Absolute Borel classes      115
Absolute multiplicative Borel class      115
Absolutely closed-monotone      78
Absolutely monotone      82
Absolutely open-monotone      110
Addition theorem      6
Additive Borel class      114
Additive in U      130
Ambiguous      231
Axioms A      178
Axioms Al      180
Axioms H      175
Axioms H extended      176
Axioms M      171
Axioms N      172
Axioms S      179
Axioms Sh      180
Base for the closed sets      251
Base-normal (base)      259
Base-normal (space)      259
Basic inductive dimension modulo $\mathcal{P}$      218
Border structure of order n      308
Borel set      112
Boundary dimension      305
Brouwer fixed point theorem      12
class      74
Closed-additive in $\mathcal{U}$      130
Closed-monotone      77
Closed-monotone in $\mathcal{U}$      77
Coincidence theorem      10
Combinatorially equivalent      88 96
Compactification problem      16
Compactness deficiency      16
Complementary      231
complete      28
Completeness deficiency      34
Completeness order dimension      40
Cosmic family in $\mathcal{U}$      142
Countable sum theorem      6
Countably closed-additive in $\mathcal{U}$      142
Cover      42
Covering completeness degree      49
Covering dimension      42
de Groot’s problem      16
de Groot’s theorem      15 280
Decomposition theorem      6
Deep inside      300
Deficiency, compactness      16
Deficiency, completeness      34
Deficiency, large $\mathcal{P}-$      109
Deficiency, small $\mathcal{P}-$      109
Degree, covering completeness      49
Degree, large compactness      21
Degree, large inductive $\sigma$-compactness      54
Degree, large inductive compactness      20
Degree, large inductive completeness      33
Degree, small inductive $\sigma$-compactness      54
Degree, small inductive compactness      15
Degree, small inductive completeness      33
Diameter      30
Dimension of a mapping      206
Dimension offset      208
Dimension, basic inductive modulo $\mathcal{P}$      218
Dimension, boundary      305
Dimension, completeness order      40
Dimension, covering      42
Dimension, large covering modulo $\mathcal{P}$      94
Dimension, large inductive      9
Dimension, large inductive modulo $\mathcal{P}$      80
Dimension, mixed inductive      241
Dimension, order      25
Dimension, order modulo $\mathcal{P}$      234
Dimension, proximity      303
Dimension, small covering modulo $\mathcal{P}$      94
Dimension, small inductive      3
Dimension, small inductive modulo $\mathcal{P}$      76
Dimensionsgrad      2
Disjunction property      251
Disjunctive      259
Distributive      248
Dowker space      182
Dowker universe      156
Dowker-open      149
Dual ideal      249
Enclosure      49 94
Equimorphism      301
Example, Kimura’s      68
Example, Pol’s      62
Example, Roy’s      9
Excision theorem      224
expansion      226
Extendable $\mathcal{K}$-border cover      304
Extension      28
FAR      300
Framework of a base      219
Freudenthal base      267
Freudenthal compactification      267
Hayashi sponge      165
Hereditary with respect to $G_{\delta}$-sets      32
Homomorphism      249
Ideal compactification      266
Inductive invariants      126
Isomorphic      249
Isomorphism      249
Join      248
Kimura’s example      68
Kimura’s theorem      68 297
Large $\sigma$-deficiency      109
Large compactness degree      21
Large covering dimension modulo V      94
Large inductive compactness degree      20
Large inductive completeness degree      33
Large inductive dimension      9
Large inductive dimension modulo $\mathcal{P}$      80
Lattice      248
Lavrentieff’s theorem      31
Lindeloef at infinity      272
Locally finite sum theorem      39
Locally finitely closed-additive in W      142
Maximal      249
Meet      248
Mixed inductive dimension      241
Modulo notation      50 87
Monotone      82
Monotone in $\mathcal{U}$      82
Multiplicative Borel class      114
NEAR      300
Normal base      259
Normal family extension      147
Normal family in $\mathcal{U}$      142
Normal universe modulo $\mathcal{P}$      99
Open cover      42
Open-monotone      110
Open-monotone in $\mathcal{U}$      110
Operation $\textbf{H}$      140
Operation $\textbf{M}$      140
Operation $\textbf{N}$      147
Operation $\textbf{S}$      151
Order dimension      25
Order dimension modulo $\mathcal{P}$      234
Order of a cover      41
Order of a mapping      206
Order offset      209
Oscillation      30
Partition      3
Perfect compactification      270
Point addition theorem      8
Point-additive in $\mathcal{U}$      130
Polish space      117
Pol’s example      62
Product theorem      7
Proximal      300
Proximity dimension      303
Proximity relation      300
Proximity space      300
Quasi-component      275
Quasi-component space      275
Rational curve      15
refinement      42
Regular curve      15
Regular family in W      142
Regularly closed      267
Regularly open      267
Relative dimension      199
Remainder      302
Rim-compact      14
Ring      259
Roy’s example      9
S-deficiency      58
S-hull      58
S-kernel      55
screening      259
Screening collection      266
Semi-normal family in U      142
Separated      84
Shrinking      42
Small $P$-deficiency      109
Small covering dimension modulo $\mathcal{U}$      94
Small inductive $cr$-compactness degree      54
Small inductive compactness degree      15
Small inductive completeness degree      33
Small inductive dimension      3
Small inductive dimension modulo $\mathcal{U}$      76
Stable value modulo $\mathcal{P}$      89
Strongly closed-additive in $\mathcal{U}$      130
Strongly hereditarily normal      150
Strongly paracompact      160
Structure theorem      40
Subspace Theorem      5
Super normal      182
Swelling      88
S—Surplus      55
Theorem Addition      6
Theorem, Brouwer fixed point      12
Theorem, coincidence      10
Theorem, countable sum      6
Theorem, de Groot’s      15 280
Theorem, decomposition      6
Theorem, excision      224
Theorem, Kimura’s      68 297
Theorem, Lavrentieff’s      31
Theorem, locally finite sum      39
Theorem, point addition      8
Theorem, product      7
Theorem, structure      40
Theorem, subspace      5
Theorem, Tumarkin’s extension      32
Topologically invariant class      74
Totally normal      147
Tumarkin’s extension theorem      32
Universe of discourse      74
Unstable value modulo $\mathcal{P}$      89
Wallman compactification      254
Wallman compactification with respect to a base      260
Wallman representation      251
Weight      262
Zero-dimension ally embedded      268
Zero-set      255
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