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.