Ⅰ．FRECHET (V)SPACES 1
1．Fréchet (V)spaces 3
2．Limit elements and derived sets 3
3．Topological equivalence of (V)spaces 4
4．Closed sets 6
5．The closure of a set 7
6．Open sets．The interior of a set 11
7．Sets dense-in-themselves．The nucleus of a set．Scattered sets 13
8．Sets closed in a given set 15
9．Separated sets．Connected sets 16
10．Images and inverse images of sets．Biuniform functions 22
11．Continuity．Continuous images 24
12．Conditions for continuity in a set 26
13．A continuous image of a connected set 27
14．Homeomorphic sets 28
15．Topological properties 31
16．Limit elements of order m．Elements of condensation．m-compact sets 33
17．Cantor＇s theorem 34
18．Topological limits of a sequence of sets 36
Ⅱ．TOPOLOGICAL SPACES 38
19．Topological spaces 38
20．Properties of derived sets 40
21．Properties of families of closed sets 43
22．Properties of closure 45
23．Examples of topological spaces 47
24．Properties of relatively closed sets 50
25．Homeomorphism in topological spaces 50
26．The border of a set．Nowhere-dense sets 52
Ⅲ．TOPOLOGICAL SPACES WITH A COUNTABLE BASIS 58
27．Topological spaces with countable bases 58
28．Hereditary separability of topological spaces with countable bases 60
29．The power of an aggregate of open sets 61
30．The countability of scattered sets 62
31．The Cantor-Bendixson theorem 63
32．The Lindelof and Borel-Lebesgue theorems 65
33．Transfinite descending sequences of closed sets 66
34．Bicompact sets 69
Ⅳ．HAUSDORFF TOPOLOGICAL SPACES SATISFYING THE FIRST AXIOM OF COUNTABILITY 72
35．Hausdorff topological spaces．The limit of a sequence．Fréchet＇s (L)class 72
36．Properties of limit elements 76
37．Properties of functions continuous in a given set 78
38．The power of the aggregate of functions continuous in a given set．Topological types 79
39．Continuous images of compact closed sets．Continua 82
40．The inverse of a function continuous in a compact closed set 85
41．The power of an aggregate of open (closed) sets 88
Ⅴ．NORMAL TOPOLOGICAL SPACES 90
42．Condition of normality 90
43．The powers of a perfect compact set and a closed compact set 92
44．Urysohn＇s lemma 95
45．The power of a connected set 97
Ⅵ．METRIC SPACES 98
46．Metric spaces 98
47．Congruence of sets．Equivalence by division 100
48．Open spheres 105
49．Continuity of the distance function 106
50．Separable metric spaces 107
51．Properties of compact sets 109
52．The diameter of a set and its properties 110
53．Properties equivalent to separability 115
54．Properties equivalent to closedness and compactness 117
55．The derived set of a compact set 119
56．Condition for connectedness．ε-chains 120
57．Hilbert space and its properties 122
58．Urysohn＇s theorem．Dimensional types 128
59．Fréchet＇s space Eω and its properties 133
60．The 0-dimensional Baire space．The Cantor set 142