# Book:George F. Simmons/Introduction to Topology and Modern Analysis

## George F. Simmons: Introduction to Topology and Modern Analysis

Published $1963$, McGraw-Hill.

### Contents

Preface
PART ONE: TOPOLOGY
Chapter One $\quad$ SETS AND FUNCTIONS
1. Sets and set inclusion
2. The algebra of sets
3. Functions
4. Products of sets
5. Partitions and equivalence relations
6. Countable sets
7. Uncountable sets
8. Partially ordered sets and lattices
Chapter Two $\quad$ METRIC SPACES
9. The definition and some examples
10. Open sets
11. Closed sets
12. Convergence, completeness, and Baire's theorem
13. Continuous mappings
14. Spaces of continuous functions
15. Euclidean and unitary spaces
Chapter Three $\quad$ TOPOLOGICAL SPACES
16. The definition and some examples
17. Elementary concepts
18. Open bases and open subbases
19. Weak topologies
20. The function algebras $\mathscr C \left({X, R}\right)$ and $\mathscr C \left({X, C}\right)$
Chapter Four $\quad$ COMPACTNESS
21. Compact spaces
22. Products of spaces
23. Tychonoff's theorem and locally compact spaces
24. Compactness for metric spaces
25. Ascoli's theorem
Chapter Five $\quad$ SEPARATION
26. $T_1$-spaces and Hausdorff spaces
27. Completely regular spaces and normal spaces
28. Urysohn's lemma and the Tietze expansion theorem
29. The Urysohn imbedding theorem
30. The Stone-Čech compactification
Chapter Six $\quad$ CONNECTEDNESS
31. Connected spaces
32. The components of a space
33. Totally disconnected spaces
34. Locally connected spaces
Chapter Seven $\quad$ APPROXIMATION
35. The Weierstrass approximation theorem
36. The Stone-Weierstrass theorem
37. Locally compact Hausdorff spaces
38. The extended Stone-Weierstrass theorem

PART TWO: OPERATORS
Chapter Eight $\quad$ ALGEBRAIC SYSTEMS
39. Groups
40. Rings
41. The structure of rings
42. Linear spaces
43. The dimension of a linear space
44. Linear transformations
45. Algebras
Chapter Nine $\quad$ BANACH SPACES
46. The definition and some examples
47. Continuous linear transformations
48. The Hahn-Banach theorem
49. The natural embedding of $N$ in $N^{**}$
50. The open mapping theorem
51. The conjugate of an operator
Chapter Ten $\quad$ HILBERT SPACES
52. The definition and some simple properties
53. Orthogonal complements
54. Orthonormal sets
55. The conjugate space $H^*$
56. The adjoint of an operator
58. Normal and unitary operators
59. Projections
Chapter Eleven $\quad$ FINITE-DIMENSIONAL SPECTRAL THEORY
60. Matrices
61. Determinants and the spectrum of an operator
62. The spectral theorem
63. A survey of the situation

PART THREE: ALGEBRAS OF OPERATORS
Chapter Twelve $\quad$ GENERAL PRELIMINARIES ON BANACH ALGEBRAS
64. The definition and some examples
65. Regular and singular elements
66. Topological divisors of zero
67. The spectrum
68. The formula for the spectral radius
Chapter Thirteen $\quad$ THE STRUCTURE OF COMMUTATIVE BANACH ALGEBRAS
70. The Gelfand mapping
71. Applications of the formula $r \left({x}\right) = \lim \left\Vert{x^n}\right\Vert^{1/n}$
72. Involutions in Banach algebras
73. The Gelfand-Neumark theorem
Chapter Fourteen $\quad$ SOME SPECIAL COMMUTATIVE BANACH ALGEBRAS
74. Ideals in $\mathscr C \left({X}\right)$ and the Banach-Stone theorem
75. The Stone-Čech compactification (continued)
76. Commitative $C^*$-algebras
APPENDICES
ONE $\quad$ Fixed point theorems and some applications to analysis
TWO $\quad$ Continuous curves and the Hahn-Mazurkiewicz theorem
THREE $\quad$ Boolean algebras, Boolean rings, and Stone's theorem
Bibliography
Index of Symbols
Subject Index