MAT 622 - Topology

Syllabus for MAT 622 - Topology

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 220 (Real Analysis) or equivalent

Overview: This course concerns 'General Topology' which can be characterized as the abstract framework in which the notion of continuity can be framed and studied. Thus topology provides the basic language and structure for a large part of pure and applied mathematics.

We will take up the following topics: Open and closed sets, continuous functions, subspaces, product and quotient topologies, connected and path connected spaces, compact and locally compact spaces, Baire category theorem, separability axioms.

Detailed Syllabus:

1. Review: Operations with infinite collections of sets, axiom of choice, Zorn's lemma, real line, metric spaces.
2. Topological Spaces: Definition and examples of topological spaces, Hausdorff property, fine and coarse topologies, subspace topology, closed sets, continuous functions, homeomorphisms, pasting lemma, product topology, quotient topology.
3. Connectedness and Compactness: Connected spaces and subsets, path connectedness, compact spaces and subsets, tube lemma, Tychonoff theorem, local compactness, one-point compactification, Baire category theorem.
4. Separation Axioms: First and second countability, separability, separation axioms (T₁ etc.), normal spaces, Urysohn lemma, Tietze extension theorem.
5. Topics for Student Presentations: Order topology, quotients of the square, locally (path) connected spaces, sequential and limit point compactness, topological groups, nets, applications of Baire category theorem.

Main Reference:

• Topology by James R. Munkres, 2nd Edition. Pearson Education, Indian Reprint, 2001.

Other References:

• Basic Topology by M. A. Armstrong. Springer-Verlag, Indian Reprint, 2004.
• Topology by K. Jänich. Undergraduate Texts in Mathematics, Springer-Verlag, 1984.
• Introduction to Topology and Modern Analysis by G. F. Simmons. International Student Edition. McGraw-Hill, Singapore, 1963.
• Topology of Metric Spaces by S. Kumaresan. 2^nd edition, Narosa, 2011.