MTG 5316 Introduction to Topology 1

  1. Set theory: Sets, functions, index sets, Cartesian products, finite and infinite sets, cardinality, Cantor-Schroeder-Bernstein Theorem.
  2. Definitions and examples of topology, basis, open and closed sets, interior, boundary, closure. Continuous maps, homeomorphism. The subspace topology, quotient spaces, product spaces. Metric topology. Complete metric spaces. The contracting mapping theorem. Baire category theorem. Separation axioms. Normal spaces. Tietze Extension Theorem.
  3. Function spaces and their topologies.
  4. Connectedness. Connectedness in the real line, path-connectedness, components, local connectivity.
  5. Compactness. Covers, finite intersection property. Sequential compactness. Compactness in the real line and in Euclidean spaces.