Detailed Table of Contents

Preface, xiii
 -- About the Choice of Topics, xiii
 -- Existence, Examples, and Intangibles, xv
 -- Abstract versus Concrete, xviii
 -- Order of Topics, xix
 -- How to Use This Book, xx
 -- Acknowledgments, xxi
 -- To Contact Me, xxii


Part A: SETS AND ORDERINGS, 1

1. Sets, 3
 -- Mathematical Language and Informal Logic, 3
 -- Basic Notations for Sets, 11
 -- Ways to Combine Sets, 15
 -- Functions and Products of Sets, 19
 -- ZF Set Theory, 25

2. Functions, 34
 -- Some Special Functions, 34
 -- Distances, 39
 -- Cardinality, 43
 -- Induction and Recursion on the Integers, 47

3. Relations and Orderings, 49
 -- Relations, 50
 -- Preordered Sets, 52
 -- More about Equivalences, 54
 -- More about Posets, 56
 -- Max, Sup, and Other Special Elements, 59
 -- Chains, 62
 -- Van Maaren's Geometry-Free Sperner Lemma, 64
 -- Well Ordered Sets, 72

4. More about Sups and Infs, 78
 -- Moore Collections and Moore Closures, 78
 -- Some Special Types of Moore Closures, 83
 -- Lattices and Completeness, 87
 -- More about Lattices, 88
 -- More about Complete Lattices, 91
 -- Order Completions, 92
 -- Sups and Infs in Metric Spaces, 97

5. Filters, Topologies, and Other Sets of Sets, 100
 -- Filters and Ideals, 100
 -- Topologies, 106
 -- Algebras and Sigma-Algebras, 115
 -- Uniformities, 118
 -- Images and Preimages of Sets of Sets, 122
 -- Transitive Sets and Ordinals, 122
 -- The Class of Ordinals, 127

6. Constructivism and Choice, 131
 -- Examples of Nonconstructive Mathematics, 132
 -- Further Comments on Constructivism, 135
 -- The Meaning of Choice, 139
 -- Variants and Consequences of Choice, 141
 -- Some Equivalents of Choice, 144
 -- Countable Choice, 148
 -- Dependent Choice, 149
 -- The Ultrafilter Principle, 150

7. Nets and Convergences, 155
 -- Nets, 157
 -- Subnets, 161
 -- Universal Nets, 165
 -- More about Subsequences, 167
 -- Convergence Spaces, 168
 -- Convergence in Posets, 171
 -- Convergence in Complete Lattices, 174


Part B: ALGEBRA, 177

8. Elementary Algebraic Systems, 179
 -- Monoids, 179
 -- Groups, 181
 -- Sums and Quotients of Groups, 184
 -- Rings and Fields, 187
 -- Matrices, 192
 -- Ordered Groups, 194
 -- Lattice Groups, 197
 -- Universal Algebras, 202
 -- Examples of Equational Varieties, 205

9. Concrete Categories, 208
 -- Definitions and Axioms, 210
 -- Examples of Categories, 212
 -- Initial Structures and Other Categorical Constructions, 217
 -- Varieties with Ideals, 221
 -- Functors, 227
 -- The Reduced Power Functor, 229
 -- Exponential (Dual) Functors, 238

10. The Real Numbers, 242
 -- Dedekind Completions of Ordered Groups, 242
 -- Ordered Fields and the Reals, 245
 -- The Hyperreal Numbers, 250
 -- Quadratic Extensions and the Complex Numbers, 254
 -- Absolute Values, 259
 -- Convergence of Sequences and Series, 263

11. Linearity, 272
 -- Linear Spaces and Linear Subspaces, 272
 -- Linear Maps, 277
 -- Linear Dependence, 280
 -- Further Results in Finite Dimensions, 282
 -- Choice and Vector Bases, 285
 -- Dimension of the Linear Dual (Optional), 287
 -- Preview of Measure and Integration, 288
 -- Ordered Vector Spaces, 292
 -- Positive Operators, 296
 -- Orthogonality in Riesz Spaces (Optional), 300

12. Convexity, 302
 -- Convex Sets, 302
 -- Combinatorial Convexity in Finite Dimensions (Optional), 307
 -- Convex Functions, 308
 -- Norms, Balanced Functionals, and Other Special Functions, 313
 -- Minkowski Functionals, 315
 -- Hahn-Banach Theorems, 317
 -- Convex Operators, 319

13. Boolean Algebras, 326
 -- Boolean Lattices, 326
 -- Boolean Homomorphisms and Subalgebras, 329
 -- Boolean Rings, 334
 -- Boolean Equivalents of UF, 338
 -- Heyting Algebras, 340

14. Logic and Intangibles, 344
 -- Some Informal Examples of Models, 345
 -- Languages and Truths, 350
 -- Ingredients of First-Order Language, 354
 -- Assumptions in First-Order Logic, 362
 -- Some Syntactic Results (Propositional Logic), 366
 -- Some Syntactic Results (Predicate Logic), 372
 -- The Semantic View, 377
 -- Soundness, Completeness, and Compactness, 385
 -- Nonstandard Analysis, 394
 -- Summary of Some Consistency Results, 399
 -- Quasiconstructivism and Intangibles, 403


Part C: TOPOLOGY AND UNIFORMITY, 407

15. Topological Spaces, 409
 -- Pretopological Spaces, 409
 -- Topological Spaces and Their Convergences, 411
 -- More about Topological Closures, 415
 -- Continuity, 417
 -- More about Initial and Product Topologies, 421
 -- Quotient Topologies, 425
 -- Neighborhood Bases and Topology Bases, 426
 -- Cluster Points, 430
 -- More about Intervals, 431

16. Separation and Regularity Axioms, 435
 -- Kolmogorov (T-Zero) Topologies and Quotients, 436
 -- Symmetric and Frechet (T-One) Topologies, 438
 -- Preregular and Hausdorff (T-Two) Topologies, 439
 -- Regular and T-Three Topologies, 441
 -- Completely Regular and Tychonov (T-Three and a Half) Topologies, 442
 -- Partitions of Unity, 444
 -- Normal Topologies, 446
 -- Paracompactness, 448
 -- Hereditary and Productive Properties, 451

17. Compactness, 453
 -- Characterizations in Terms of Convergences, 453
 -- Basic Properties of Compactness, 456
 -- Regularity and Compactness, 458
 -- Tychonov's Theorem, 461
 -- Compactness and Choice (Optional), 461
 -- Compactness, Maxima, and Sequences, 466
 -- Pathological Examples: Ordinal Spaces (Optional), 472
 -- Boolean Spaces, 473
 -- Eberlein-Smulian Theorem, 477

18. Uniform Spaces, 481
 -- Lipschitz Mappings, 482
 -- Uniform Continuity, 484
 -- Pseudometrizable Gauges, 487
 -- Compactness and Uniformity, 490
 -- Uniform Convergence, 491
 -- Equicontinuity, 493

19. Metric and Uniform Completeness, 499
 -- Cauchy Filters, Nets, and Sequences, 499
 -- Complete Metrics and Uniformities, 502
 -- Total Boundedness and Precompactness, 505
 -- Bounded Variation, 508
 -- Cauchy Continuity, 511
 -- Cauchy Spaces (Optional), 512
 -- Completions, 513
 -- Banach's Fixed Point Theorem, 516
 -- Meyers's Converse (Optional), 520
 -- Bessaga's Converse and Bronsted's Principle (Optional), 523

20. Baire Theory, 530
 -- G-Delta Sets, 530
 -- Meager Sets, 531
 -- Generic Continuity Theorems, 533
 -- Topological Completeness, 536
 -- Baire Spaces and the Baire Category Theorem, 537
 -- Almost Open Sets, 539
 -- Relativization, 540
 -- Almost Homeomorphisms, 541
 -- Tail Sets, 543
 -- Baire Sets (Optional), 545

21. Positive Measure and Integration, 547
 -- Measurable Functions, 547
 -- Joint Measurability, 549
 -- Positive Measures and Charges, 552
 -- Null Sets, 554
 -- Lebesgue Measure, 556
 -- Some Countability Arguments, 559
 -- Convergence in Measure, 561
 -- Integration of Positive Functions, 565
 -- Essential Suprema, 569


Part D: TOPOLOGICAL VECTOR SPACES, 573

22. Norms, 575
 -- (G-)(Semi)Norms, 575
 -- Basic Examples, 578
 -- Sup Norms, 581
 -- Convergent Series, 585
 -- Bochner-Lebesgue Spaces, 589
 -- Strict Convexity and Uniform Convexity, 596
 -- Hilbert Spaces, 601

23. Normed Operators, 607
 -- Norms of Operators, 607
 -- Equicontinuity and Joint Continuity, 612
 -- The Bochner Integral, 65
 -- Hahn-Banach Theorems in Normed Spaces, 617
 -- A Few Consequences of HB, 621
 -- Duality and Separability, 622
 -- Unconditionally Convergent Series, 624
 -- Neumann Series and Spectral Radius (Optional), 627

24. Generalized Riemann Integrals, 629
 -- Definitions of the Integrals, 629
 -- Basic Properties of Gauge Integrals, 635
 -- Additivity over Partitions, 638
 -- Integrals of Continuous Functions, 642
 -- Monotone Convergence Theorem, 645
 -- Absolute Integrability, 647
 -- Henstock and Lebesgue Integrals, 649
 -- More about Lebesgue Measure, 656
 -- More about Riemann Integrals (Optional), 658

25. Frechet Derivatives, 661
 -- Definitions and Basic Properties, 661
 -- Partial Derivatives, 665
 -- Strong Derivatives, 669
 -- Derivatives of Integrals, 674
 -- Integrals of Derivatives, 675
 -- Some Applications of the Second Fundamental Theorem of Calculus, 677
 -- Path Integrals and Analytic Functions (Optional), 683

26. Metrization of Groups and Vector Spaces, 688
 -- F-Seminorms, 689
 -- TAG's and TVS's, 697
 -- Arithmetic in TAG's and TVS's, 700
 -- Neighborhoods of Zero, 702
 -- Characterizations in Terms of Gauges, 705
 -- Uniform Structure of TAG's, 708
 -- Pontryagin Duality and Haar Measure (Optional; Proofs Omitted), 710
 -- Ordered Topological Vector Spaces, 714

27. Barrels and Other Features of TVS's, 721
 -- Bounded Subsets of TVS's, 721
 -- Bounded Sets in Ordered TVS's, 726
 -- Dimension in TVS's, 728
 -- Fixed Point Theorems of Brouwer, Schauder, and Tychonov, 730
 -- Barrels and Ultrabarrels, 732
 -- Proofs of Barrel Theorems, 736
 -- Inductive Topologies and LF Spaces, 744
 -- The Dream Universe of Garnir and Wright, 748

28. Duality and Weak Compactness, 752
 -- Hahn-Banach Theorems in TVS's, 752
 -- Bilinear Pairings, 754
 -- Weak Topologies, 758
 -- Weak Topologies of Normed Spaces, 761
 -- Polar Arithmetic and Equicontinuous Sets, 764
 -- Duals of Product Spaces, 769
 -- Characterizations of Weak Compactness, 771
 -- Some Consequences in Banach Spaces, 777
 -- More about Uniform Convexity, 780
 -- Duals of the Lebesgue Spaces, 782

29. Vector Measures, 785
 -- Basic Properties, 785
 -- The Variation of a Charge, 787
 -- Indefinite Bochner Integrals and Radon-Nikodym Derivatives, 790
 -- Conditional Expectations and Martingales, 792
 -- Existence of Radon-Nikodym Derivatives, 796
 -- Semivariation and Bartle Integrals, 802
 -- Measures on Intervals, 806
 -- Pincus's Pathology (Optional), 810

30. Initial Value Problems, 814
 -- Elementary Pathological Examples, 815
 -- Caratheodory Solutions, 816
 -- Lipschitz Conditions, 819
 -- Generic Solvability, 822
 -- Compactness Conditions, 822
 -- Isotonicity Conditions, 824
 -- Generalized Solutions, 826
 -- Semigroups and Dissipative Operators, 828

References, 839
Index and Symbol List, 857