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LIST OF SYMBOLS 5

INTRUODUCTION 9

1．(Generalized) topological spaces 13

2．Neighborhood spaces 15

3．Accumulation points;interior,exterior and boundary of a set;closed sets and open sets 18

Characterization of neighborhood spaces by means of a generalized topology 19

Neighborhood spaces whose neighborhood bases consist of open neighborhoods 21

4．Topological spaces 22

5．Topological base 31

6．Continuous mappings of a space into a space;homeomorphisms 34

7．Topologization of a set of by a mapping 41

Subspaces 45

Topological product 48

Topological quotient 52

8．Separation axioms;classification of spaces by means of the separation axioms 52

9．Topologization of a set by means of a real distance;metric spaces 62

Uniformly continuous mappings of a metric space into a metric space 70

Several examples of metric spaces 73

Geodesic metric 74

Normed linear,or vector,spaces 76

Metrization of topological spaces 77

10．Topologizatoin of a set by the uniform topology;Efremovich＇s proximity spaces or δ-spaces 85

Mappings of a δ-space into a δ-space;δ-functions;δ-homeomorphisms (equimorphisms) 96

Metrization of a δ-space 98

11．Topologization of a set by means of the uniform topology;Weil uniform spaces 100

Mappings of uniform spaces into uniform spaces;μ-functions and uniform homeomorphisms 109

Uniformizable spaces 110

Metrization of uniform spaces 115

Tukey-Smirnov uniformities;relation between δ-spaces and uniform spaces 116

Shvarts uniformities 119

12．Topologization of a set by means of an abstract distance 121

Abstract distance and neighborhood spaces 121

Abstract distance and topological spaces 124

Abstract distance and uniform spaces 126

The Kurepa class of spaces E [M] 128

Spaces I (Wa) 131

R-spaces 132

eT-spaces 136

Continuous mappings of spaces defined by an abstract distance 138

Topologization of a set by means of a partially ordered distance 140

Partially ordered distance in the sense of Appert＇s definition 140

Colmez partially ordered distance 142

Topologization of a set by a totally ordered distance;pseuo-distancial,or Kurepa-Fréchet,spaces 146

The Kurepa class of spaces (Da) 149

LITERATURE 151

INDEX 157