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Sets, Functions, and Relations | |

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Sets and Membership | |

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Some Remarks on the Use of the Connectives and, or, implies | |

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Subsets | |

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Union and Intersection of Sets | |

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Complementation | |

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Set Identities and Other Set Relations | |

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Counterexamples | |

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Collections of Sets | |

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Cartesian Product | |

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Functions | |

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Relations | |

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Set Inclusions for Image and Inverse Image Seta | |

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The Restriction of a Function | |

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Composition of Functions | |

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Sequences | |

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Subsequences | |

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Finite Induction and Well-Ordering for Positive Integers | |

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Sequencea Defined Inductively | |

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Some Important Properties of Relations | |

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Decomposition of a Set | |

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Equivalence Classes | |

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Partially Ordered and Totally Ordered Seta | |

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Properties of Boundedness for Partially Ordered Sets | |

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Axiom of Choice and Zorn's Lemma | |

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Cardinality of Sets (Introduction) | |

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Countable Sets | |

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Uncountable Sets | |

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Nonequivalent Sets | |

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Review Exercises | |

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Structure Of R and Rn | |

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Algebraic Structure of R | |

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Distance Between Two Points in R | |

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Limit of a Sequence in R | |

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The Nested Interval Theorem for R | |

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Algebraic Structure for R<sup>n</sup> | |

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The Cauchy-Schwarz Inequality | |

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The Distance Formula in R<sup>n</sup> | |

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Open Subsets of R<sup>n</sup> | |

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Limit Points in R<sup>n</sup> | |

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Closed Subsets of R<sup>n</sup> | |

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Bounded Subsets of R | |

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Convergent Sequences in R<sup>n</sup> | |

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Cauchy Criterion for Convergence | |

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Some Additional Properties for R<sup>n</sup> | |

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Some Further Remarks about R<sup>n</sup> | |

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Metric Spaces: Introduction | |

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Distance Function and Metric Spaces | |

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Open Sets and Closed Sets | |

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Some Basic Theorems Concerning Open and Closed Sets | |

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Topology Generated by a Metric | |

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Subspace of a Metric Space | |

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Convergent Sequences in Metric Spaces | |

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Cartesian Product of a Finite Number of Metric Spaces | |

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Continuous Mappings: Introduction | |

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Uniform Continuity | |

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Metric Spaces: Special Properties and Mappings on Metric Spaces | |

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Separation Properties | |

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Connectedness in Metric Spaces | |

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The Invariance of Connectedness under Continuous Mappings | |

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Polygonal Connectedness | |

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Separable Metric Spaces | |

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Totally Bounded Metric Spaces | |

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Sequential Compactness for Metric Spaces | |

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The Bolzano-Weierstrass Property | |

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Compactness or Finite Subcovering Property | |

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Complete Metric Spaces | |

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Nested Sequences of Sets for Complete Spaces | |

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Another Characterization of Compact Metric Spaces | |

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Completion of a Metric Space | |

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Sequences of Mappings into a Metric Space | |

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Review Exercises | |

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Metric Spaces: Some Examples and Applications | |

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Linear or Vector Spaces | |

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The Hilbert Space l<sup>2</sup> | |

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The Hilbert Cube | |

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The Space & ([a, b]) of Continuous Real-Valued Mappings on a Closed Interval [a, b] | |

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An Application of Completeness: Contraction Mappings | |

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Fundamental Existence Theorem for First Order Differential Equations-An Application of the Banach Fixed Point Theorem | |

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General Topological Spaces and Mappings on Topological Spaces | |

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Topological Spaces | |

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Base for a Topology | |

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Some Basic Definitions | |

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Some Basic Theorems for Topological Spaces | |

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Neighborhoods and Neighborhood Systems | |

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Subspaces | |

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Continuous and Topological Mappings | |

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Some Basic Theorems Concerning Mappings | |

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Separation Properties for Topological Spaces | |

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A Characterization of Normality | |

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Separability Axiom | |

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Second Countable Spaces | |

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First Countable Spaces | |

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Comparison of Topologies | |

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Urysohn's Metrization Theorem | |

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Compactness And Related Properties | |

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Definitions of Various Compactness Properties | |

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Some Consequences of Compactness | |

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Relations Between Various Types of Compactness | |

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Local Compactness | |

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The One-Point Compactification | |

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Some Generalizations of Mappings Denned on Compact Spaces | |

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Connectedness And Related Concepts | |

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Connectedness. Definitions | |

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Some Basic Theorems Concerning Connectedness | |

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Limit Superior and Limit Inferior of Sequences of Subsets of a Space | |

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Review Exercises | |

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Quotient Spaces | |

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Decomposition of a Topological Space | |

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Quasi-Compact Mappings | |

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The Quotient Topology | |

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Decomposition of a Domain Space into Point Inverses | |

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Topologically Equivalent Mappings | |

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Decomposition of a Domain Space into Components of Point Inverses | |

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Factorization of Compact Mappings | |

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Net and Filter Convergence | |

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Nets and Subnets | |

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Convergence of Nets | |

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Filters | |

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Product Spaces | |

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Cartesian Products | |

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The Product Topology | |

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Mappings into Product Spaces | |

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References | |

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Index | |