MATH 423/523 - INTRO TO REAL ANALYSIS
Important Ideas, Terms, and Topics
Latest revision: 11/19/08

Chapter 1: The Real and Complex Number Systems.
  1. Field axioms: commutativity, associativity, distributivity, identities, and inverses.
  2. Order axioms. The real line, open and closed intervals, interval notation.
  3. Supremums and infimums, proofs involving sup’s and inf’s.
  4. The completeness axiom and related axioms, Cauchy sequences, the Archimedean axiom.
  5. Inductive sets and the positive integers. Unique factorization. Greatest common divisors.
  6. Rational and irrational numbers. Decimal expansions. 0.999... vs. 1.
  7. Rn and the Cauchy-Schwarz and triangle inequalities.
  8. Complex numbers: equality, sums, products, differences, quotients, absolute values, conjugates, arguments.
  9. The geometry of complex numbers, polar representation, complex exponentials, DeMoivre’s theorem.

Chapter 2: Some Basic Notions of Set Theory.
  1. Sets, subsets, and the empty set. Products, unions, intersections, and differences of sets.
  2. Relations on sets and between sets. Equivalence relations, reflexive, symmetric, and transitive relations.
  3. Functions, one-to-one functions, onto functions. Images and inverse images. Compositions. Inverses, left and right inverses.
  4. Cardinality, equinumerous sets. Finite, countable, and uncountable sets.
  5. Proofs involving cardinality. Countability and sequences. Finite products and countable unions of countable sets.
  6. Using injections and surjections instead of bijections to prove countability.
  7. Proof techniques: direct proofs, proofs by contrapositive and contradiction, proofs by cases. Quantifiers, negations.

Chapter 3: Elements of Point Set Topology.
  1. Euclidean space Rn: vector space, inner product space, and Euclidean norm properties.
  2. General norms on vector spaces: positivity, scaling, triangle inequality.
  3. Metric spaces, properties of metrics, metrics defined by norms, examples.
  4. Open and closed balls, interior points, open sets.
  5. General topologies, open sets, countable unions, finite intersections.
  6. Open sets in R: countable unions of disjoint open intervals.
  7. Closed sets, adherent and accumulation points, characterizations of closed sets.
  8. Interiors, closures, boundaries, and derived sets. Extremal aspects of interiors and closures.
  9. The Bolzano-Weierstrass Theorem, existence of accumulation points, the Cantor Intersection Theorem.
  10. Compactness, open covers, The Heine-Borel Theorem, compactness in Rn.
  11. Subspaces of metric spaces, relative topologies. Dense subsets.

Chapter 4: Limits and Continuity.
  1. Limits of sequences in metric spaces. Adherent and accumulation points as limits of sequences.
  2. Cauchy sequences, complete metric spaces, compact implies complete.
  3. Limits of functions between metric spaces, properties of limits with complex and vector valued functions.
  4. Continuity of functions, continuity at a point, continuity on a set. Continuity of complex and vector valued functions.
  5. Continuity and inverse images of open and/or closed sets. Continuity of compositions.
  6. Continuous functions on compact sets. Isometry and homeomorphism.

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Latest revision: 11/19/2008