Topology: Handwritten Notesby

Tahir Mehmood

Partial Contents

These are the handwritten notes. These notes are lecture delivered by Mr. Tahir Mehmood.

1. Metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

2. Minkowski’s inequality . . . . . . . . . . . . . . . . . . .5

3. Open set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4. Closed ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5. Closed set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6. Bounded set . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

7. Limit point . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

8. Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . .14

9. Convergence in metric space and completemetric space . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

10. Cauchy sequence . . . . . . . . . . . . . . . . . . . . . . . 19

11. Bounded sequence . . . . . . . . . . . . . . . . . . . . . . 20

12. Nested interval property or Cantor’s inter-section theorem . . . . . . . . . . . . . . . . . . . . . . . . 26

13. Continuous function . . . . . . . . . . . . . . . . . . . . 28

14. Topological spaces . . . . . . . . . . . . . . . . . . . . . . 38

15. Metric topology, cofinite topology . . . . . . 39

16. Open set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

17. Closed set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

18. Closure of a set . . . . . . . . . . . . . . . . . . . . . . . . .44

19. Neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . . 48

20. Interior point, exterior point . . . . . . . . . . . .49

21. Boundary point . . . . . . . . . . . . . . . . . . . . . . . . 50

22. Limit point (with respect to topology) . . 52

23. Isolated point . . . . . . . . . . . . . . . . . . . . . . . . . . 62

24. Dense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

25. Separable set; Countable set . . . . . . . . . . . . 64

26. Base of topology . . . . . . . . . . . . . . . . . . . . . . . 65

27. Neighbourhood base or local base or base ata point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

28. Open cover; Lindelof space . . . . . . . . . . . . . 73

29. Lindelof theorem . . . . . . . . . . . . . . . . . . . . . . . 74

30. Relative topology, subspace . . . . . . . . . . . . .77

31. Separation axioms; T0-space . . . . . . . . . . . . 85

32. T1-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87

33. Subbase; Generation of topologies . . . . . . 92

34. T2-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93

35. Continuous function (with respect to topolo-gies) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

36. Product topology . . . . . . . . . . . . . . . . . . . . . . .98

37. Convergence of sequence in topologicalspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101

38. Regular space . . . . . . . . . . . . . . . . . . . . . . . . . 109

39. Completely regular space . . . . . . . . . . . . . . 111

40. Compactness in topological spaces . . . . .125

41. Homeomorphism . . . . . . . . . . . . . . . . . . . . . . 134

42. Countably compact space . . . . . . . . . . . . . 141

43. Bolzano Weierstrass property . . . . . . . . . .145

44. Lebesgue number; Big set; Lebesgue coverylemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

45. ε−net; Totally bounded . . . . . . . . . . . . . . . 149

46. Connected spaces; Disconnected . . . . . . . 157

47. Component . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

48. Totally disconnected . . . . . . . . . . . . . . . . . . 173

49. Separated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

50. Normed spaced . . . . . . . . . . . . . . . . . . . . . . . .186

51. Uniformly continuous . . . . . . . . . . . . . . . . . 189

52. Closed unit ball; Convex set . . . . . . . . . . . 190

53. Vector space . . . . . . . . . . . . . . . . . . . . . . . . . . 191

54. Linear combination; Spanning set; Linearlyindependent . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

55. Linearly dependent . . . . . . . . . . . . . . . . . . . . 193

56. Linearly independent lemma . . . . . . . . . . 194

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57. Finite dimensional; Subspace . . . . . . . . . . 197

58. Equivalent norms . . . . . . . . . . . . . . . . . . . . . 200

59. Banach space . . . . . . . . . . . . . . . . . . . . . . . . . 205

60. Reiz Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 222

61. Hilbert spaces; Inner product spaces . . . 224

62. Polarization identity . . . . . . . . . . . . . . . . . . 228

63. Cauchy Schewarz inequality . . . . . . . . . . . 229

64. Appalonius identity . . . . . . . . . . . . . . . . . . . 231

65. Hilbert space; Pythagorian theorem . . . 233

66. Minimizing vector . . . . . . . . . . . . . . . . . . . . . 235

67. Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

68. Orthogonal set; Orthonormal set . . . . . . 242

69. Bessel’s inequality . . . . . . . . . . . . . . . . . . . . . 243

70. Total orthonormal sets (definition); Parse-vel’s equality . . . . . . . . . . . . . . . . . . . . . . . . . . 245

71. Linear Operator; The Kernel or Null spaceof a linear operator; Continuous linear oper-ator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

72. Bounded linear operator . . . . . . . . . . . . . . .249

73. Norm of a bounded lienar operator . . . . 252

74. Linear functionals . . . . . . . . . . . . . . . . . . . . . 260

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