Mathematics Honors Courses in Armenia

Initiative of Armenian Society of Fellows (ASOF)

Metric Spaces and Topology

2023 Spring

class

Lecture info

Problem session info

Details

 tree

Cantor space

Lectures

Homework

Prerequisits

 Comfort with definitions and proofs, as well as familiarity with basic real analysis (e.g. rigorous understanding of limits of sequences of reals).

Coursework and grades

Course description

 Metric spaces are sets with a given metric, i.e. distance function. For example, ℝ with the distance between points x, y being |x-y|, or the set C([0,1]) of continuous functions on [0,1] with the distance between functions f, g being supx ∈ [0,1] |f(x) - g(x)|. Metric measures how different two objects are and it can be used to understand how objects are located with respect to others. Most importantly, metric yields a notion of convergence, which allows for proving the existence of a desirable object (e.g. a solution to a PDE) by only building approximations to it.
 Besides analysis on metric spaces, we will also study their generalization, namely, topological spaces, where the notion of closeness need not satisfy the "triangle inequality." Social proximity is such an example: your parents know you and you know your friends, but your parents may not know your friends.

Topics

  • Open sets in metric spaces and convergence
  • Complete metric spaces and completion
  • Separability and second countability
  • Continuity and uniform convergence
  • Basic set theory: equinumerosity and countability
  • Cardinality of metric spaces and the perfect set property
  • Baire category theorem, Baire measurability, and applications
  • Topological spaces
  • Bases and generation, product topology
  • Connectedness and separation axioms
  • Continuous functions and extensions
  • Compactness and Tychonoff's theorem
  • Sequential compactness and first countability
  • Compactness for metric spaces
  • Nets and compactness in terms of nets

    • Textbooks

  • I. KAPLANSKY, Set Theory and Metric Spaces (2nd ed.), AMS Chelsea Publishing, 1957.
  • T. GAMELIN, R. GREENE, Introduction to topology (2nd ed.), Mineola, N.Y., Dover Publications, 1999.
  • G. FOLLAND, Real Analysis: Modern Techniques and Their Applications (2nd ed.), Pure and Applied Mathematics (N.Y.), John Wiley & Sons, 1999.