General TopologyPDF电子书下载

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