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Tables of Contents for General Topology
Chapter/Section Title
Page #
Page Count
I. FRECHET (V)SPACES
Frechet (V)spaces
3
1
Limit elements and derived sets
3
1
Topological equivalence of (V)spaces
4
2
Closed sets
6
1
The closure of a set
7
4
Open sets. The interior of a set
11
2
Sets dense-in-themselves. The nucleus of a set. Scattered sets
13
2
Sets closed in a given set
15
1
Separated sets. Connected sets
16
6
Images and inverse images of sets. Biuniform functions
22
2
Continuity. Continuous images
24
2
Conditions for continuity in a set
26
1
A continuous image of a connected set
27
1
Homeomorphic sets
28
3
Topological properties
31
2
Limit elements of order m. Elements of condensation. m-compact sets
33
1
Cantor's theorem
34
2
Topological limits of a sequence of sets
36
2
II. TOPOLOGICAL SPACES
Topological spaces
38
2
Properties of derived sets
40
3
Properties of families of closed sets
43
2
Properties of closure
45
2
Examples of topological spaces
47
3
Properties of relatively closed sets
50
1
Homeomorphism in topological spaces
50
2
The border of a set. Nowhere-dense sets
52
6
III. TOPOLOGICAL SPACES WITH A COUNTABLE BASIS
Topological spaces with countable bases
58
2
Hereditary separability of topological spaces with countable bases
60
1
The power of an aggregate of open sets
61
1
The countability of scattered sets
62
1
The Cantor-Bendixson theorem
63
2
The Lindelof and Borel-Lebesgue theorems
65
1
Transfinite descending sequences of closed sets
66
3
Bicompact sets
69
3
IV. HAUSDORFF TOPOLOGICAL SPACES SATISFYING THE FIRST AXIOM OF COUNTABILITY
Hausdorff topological spaces. The limit of a sequence. Frechet's (L)class
72
4
Properties of limit elements
76
2
Properties of functions continuous in a given set
78
1
The power of the aggregate of functions continuous in a given set. Topological types
79
3
Continuous images of compact closed sets. Continua
82
3
The inverse of a function continuous in a compact closed set
85
3
The power of an aggregate of open (closed) sets
88
2
V. NORMAL TOPOLOGICAL SPACES
Condition of normality
90
2
The powers of a perfect compact set and a closed compact set
92
3
Urysohn's lemma
95
2
The power of a connected set
97
1
VI. METRIC SPACES
Metric spaces
98
2
Congruence of sets. Equivalence by division
100
5
Open spheres
105
1
Continuity of the distance function
106
1
Separable metric spaces
107
2
Properties of compact sets
109
1
The diameter of a set and its properties
110
5
Properties equivalent to separability
115
2
Properties equivalent to closedness and compactness
117
2
The derived set of a compact set
119
1
Condition for connectedness. &epsis;-chains
120
2
Hilbert space and its properties
122
6
Urysohn's theorem. Dimensional types
128
5
Frechet's space Eω and its properties
133
9
The 0-dimensional Baire space. The Cantor set
142
4
Closed and compact sets as continuous images of the Cantor set
146
5
Biuniform and continuous images of sets
151
2
Uniform continuity
153
2
Uniform convergence of a sequence of functions
155
3
The (C) space of all functions continuous in the interval [0, 1]
158
5
The space of all bounded closed sets of a metric space
163
3
Sets Fσ and Gδ
166
1
The straight line as the sum of ℵ1 ascending sets Gδ
167
7
Hausdorff's sets Pα and Qα
174
4
Sets which are locally Pα and Qα
178
3
Sets locally of the first category
181
2
Oscillation of a function
183
3
VII. COMPLETE SPACES
Complete spaces
186
6
The complete space containing a given metric space
192
3
Absolutely closed spaces. Complete topological spaces
195
2
The category of a complete space
197
1
Continuity extended to a set Gδ
198
2
Lavrentieff's theorem
200
2
Conclusions from Lavrentieff's theorem
202
2
Topological invariance of sets Pα and Qα
204
1
Borel sets: their topological invariance
205
2
Analytic sets: defining systems
207
1
The operation A. Lusin's sieve
208
2
Fundamental properties of the operation A
210
3
Every Borel set an analytic set
213
1
Regular definin system
214
2
Condition for a set to be analytic
216
3
Continuous images of analytic sets. Topological invariance of their complements
219
1
A new condition for a set to be analytic
220
3
Generalized sieves
223
1
The power of an anlytic set
224
4
Souslin's theorem
228
4
Biuniform and continuous images of the set of irrational numbers
232
3
The property of compact closed or compact open subsets of a metric space
235
1
The theorem of Mazurkiewicz about linear sets Gδ
236
4
Biuniform and continuous images of Borel sets
240
4
The analytic set as a sum or an intersection of ℵ1 Borel sets
244
4
Projection of closed sets
248
1
Analytic sets as projections of sets Gδ
249
1
Projective seta
250
1
Universal sets
251
1
Universal sets Pn and Cn
252
1
The existence of projective sets of any given class
253
1
The universal set Fσδ
254
2
Appendix
256
23
Notes
279
10
Index
289