Topology

Most of the topics are covered in the undergraduate course MATE 4000, Introduction to Topology and the graduate course MATE 6540, Topology.
Topics:

  • Topological spaces: closed and open sets, interior and boundary of a set of a topological space. Subspace topology. Bases and sub-bases of a topology. Order and product topology.
  • Continuous functions. Homeomorphisms.
  • Metric spaces. Sequences, convergent sequences, Cauchy sequences, complete spaces.
  • Quotient topology, Initial and Final topology.
  • Connected spaces. Connected sets in R. Connected components and paths.
  • Compact spaces. Compact sets in R. limit point compactness, Stone-Cěch compactification.
  • Separation and enumeration axioms.
  • Urysohn Lemma, Tychonoff and Tietze’s Theorems.
  • Regular and completely regular spaces. Normal spaces.


Bibliography:

  • Topology, a first course, J. R. Munkres, Prentice-Hall INC., 1975.
  • General Topology, S. Willard, Addison-Wesley, Reading, Mass. 1968.
  • Topology, J. Dugundji, Allyn and Bacon, Inc., Boston, 1968.
  • General Topology, J. L. kelley, Van Nostrand, princeton, 1955.
  • Topological Spaces, E. Cěch, Rev. Ed. (transl.), Interscience, New York, 1966.