Syllabus (MATHEMATICS)

Course Type: MAJ-14

Semester: 8

Course Code: BMTMMAJ14T

Course Title: General Topology

(L-P-Tu): 4-0-0

Credit: 4

Practical/Theory: Theory

Course Objective: To be done.

Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: Upon successful completion of this course students will, CO1: Understand terms, definitions and theorems related to topology. CO2: Demonstrate knowledge and understanding of concep

Syllabus:

General Topology [Credit: 4, 60L]

• Countable and Uncountable Sets, Schroeder-Bernstein Theorem, Cantor’s Theorem. Cardinal Numbers and Cardinal Arithmetic. Continuum Hypothesis, Zorns Lemma, Axiom of Choice.
  Well-Ordered Sets, Hausdorff’s Maximal Principle. Ordinal Numbers.
• Fundamentals of Topological Spaces: Topological spaces. Bases and sub-bases. Closure & interior; exterior, boundary, accumulation points, derived sets, dense set, Gδ and Fσ sets. Neighbourhood system. Order Topology. Discrete space.
• Subspace Topology: Subspace topology and its properties; Alternative way of defining a topology using Kuratowski closure operator, interior operator and neighbourhood systems; Continuous Functions, Open maps, Closed maps and Homeomorphisms, topological property, metric topology.
• Product Spaces: Product and box topology, Projection maps. Quotient topology. Tychonoff theorem. Separation axioms, Countability axioms and Connectedness in product spaces.
• Countability Axioms: First and Second countability axioms, Separability and Lindeloffness. Characterizations of accumulation points, closed sets, open sets in a First countable space w.r.t. sequences. Heine’s continuity criterion.
• Separation Axioms: Ti spaces (i = 0, 1, 2, 3, 3½, 4, 5), their characterizations and basic properties. Urysohn’s lemma and Tieze’s extension theorem (statement only) and their applications.
• Compactness: Compactness and its basic properties, Alexander sub-base theorem, Continuous functions and compact sets. Compactness of R. Sequential compactness, BW Compactness and countable compactness. Lebesgue Number. Local compactness, compactness in metric space, totally bounded space, Arzela-Ascoli theorem.
• Connectedness: Connected and disconnected spaces, Path Connected Spaces, Connected Sets in R, Rn(n>1), Local connectedness, Components and Path Components, Totally disconnected.

Reading References:

1. John L. Kelley; General Topology; D. Van Nostrand Company, Inc.; 1995.
2. Stephen Willard; General Topology; Addison Wesley Publishing Company; 1970.
3. James Dugundji; Topology; Allyn and Bacon, Inc.; 1978.
4. James R. Munkres, Topology: A first course; Prentice Hall, India; 1974.
5. G.F. Simmons; Introduction to Topology and Modern Analysis; McGraw Hill.
6. K.D. Joshi; Introduction to General Topology; Wiley Eastern Ltd.
7. Engelking; General Topology; Polish Scientific Publishers, Warszawa.
8. L.A. Steen and J.A. Seebach; Counterexamples in Topology; Dover Publication, Inc. New York.
9. B.C. Chatterjee, M. R. Adhikari and S. Ganguly; A text book of Topology; Asian Books; 2002.
10. W.J. Thron; Topological Structures; Holt, Rinehart and Winston, Inc., New York, 1966.