Unit 1

Topological spaces, Definition and examples, Interior, Closure and Boundary, Basis and Sub-basis, Continuity, Topological Equivalence, Subspaces.

Unit 2

Connected Spaces, Theorems on Connectedness, Connected subsets of Real line, Applications of Connectedness, Path Connected Spaces.

Unit 3

Compact spaces, Compactness and Continuity, Properties of Compact Spaces, One-Point Compactification.

Unit 4

Finite and arbitrary Products, Tychnoff’s theorem, Comparison of topologies, Quotient Spaces, T0, T1 and T2 Spaces, Regular Spaces, Normal Spaces, Separation by Continuous functions.

Unit 5

Urysohn’s Lemma and Tietze Extension Theorem, Nets, Filters and Convergence, Tychnoff’s Theorem.

Course outcomes

CO1:To introduce the concept of Metric spaces as a generalization of the analysis on the real line at a level and depth appropriate for introducing Topological spaces

CO2: providing a prerequisite for the forth coming courses like Differential Geometry, Functional Analysis, Complex Analysis etc..

CO3:To introduce the student to what it means to do mathematics, as opposed to learning about mathematics or to learning to do computational exercises.

CO4:To help the student learn how to write mathematical text according to the standards of the profession.

CO5: Inspiring them to study higher-level mathematics and to become a professional mathematician

Textbook:

• Fred H. Croom, Principles of Topology, Cengage Learning.
• J.R. Munkres, Topology, PHI.
• Sidney A. Morris, Topology without Tears, EBook.

References:

1. K.D. Joshi, Introduction to General Topology, New Age Publishers. Sheldon W Davis, Topology, Tata McGraw-Hill.
2. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill.