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Professor Medynets'

Exams


Homework

  1. HW 1. Download pdf. Due: Friday, August/29.
  2. HW 2. Download pdf. Due: Friday, September/05. Solutions
  3. HW 3. Download pdf. Due: Friday, September/12.
  4. HW 4. Download pdf. Due: Friday, September/29.
  5. HW 5. Download pdf. Due: Friday, October/17.
  6. HW 6. Download pdf. Due: Monday, November/03.

Syllabus

 Introduction.

  • Sequential and Weierstrass definitions of continuity for real-valued functions. 

Metric Spaces

  • Definition of metric spaces.
  • Examples of Metric Spaces: R^n, Cauchy-Schwartz inequality, Space of functions on a closed interval, space of 0-1 sequences.
  • Open/Closed sets in Metric Spaces.
  • The structure of sets in metric spaces: interior, closure, boundary.
  • Continuous functions in metric spaces.
  • Completeness, The Baire Category Theorem.
  • The contraction mapping theorem.
  • Applications: The Perron-Frobenius Theorem.

Topological Spaces

  • Axioms of Topological Spaces. Examples. 
  • Open sets, Closed sets, Limit Points, Boundary, Closure, Interior.
  • Base of Topology.
  • Product Topology.
  • Connected Spaces. Path-Connected Spaces. Cut points.
  • Compact spaces. Heine-Borel Theorem.
  • Homeomorphisms and homeomorphic spaces.
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