METRIC SPACES (BAI405B)

METRIC SPACES

Course Code BAI405B 
CIE Marks 50
Teaching Hours/Week (L:T:P: S) 2:2:0:0 
SEE Marks 50
Total Hours of Pedagogy 40 
Total Marks 100
Credits 03 
Exam Hours 03
Examination type (SEE) Theory

Module-1: Theory of Sets

Finite and infinite sets, countable and uncountable sets, cardinality of sets, Schroder-Bernstein

theorem, cantor’s theorem, Order relation in cardinal numbers, Arithmetic of cardinal

numbers, Partially ordered set, Zorn’s lemma and axioms of choice, various set-theoretic

paradoxes.

 (8 hours)

(RBT Levels: L1, L2 and L3)

Module-2: Concepts in Metric Spaces

Definition and examples of metric spaces, Open spheres and Closed spheres, Neighborhoods,

Open sets, Interior, Exterior and boundary points, Closed sets, Limit points and isolated points,

Interior and closure of a set, Boundary of a set, Bounded sets, Distance between two sets,

Diameter of a set. (8 hours)

(RBT Levels: L1, L2 and L3)

Module-3: Complete Metric Spaces and Continuous Functions

Cauchy and Convergent sequences, Completeness of metric spaces, Cantor’s intersection

theorem, Dense sets and separable spaces, Nowhere dense sets and Baire’s category theorem,

continuous and uniformly continuous functions, Homeomorphism. Banach contraction

principle. (8 hours)

(RBT Levels: L1, L2 and L3)

Module-4: Compactness

Compact spaces, Sequential compactness, Bolzano-Weierstrass property, Compactness and finite

intersection property, Heine-Borel theorem, Totally bounded set, equivalence of compactness and

sequential compactness. (8 hours)

 (RBT Levels: L1, L2 and L3)

Module-5: Connectedness

Separated sets, Disconnected and connected sets, components, connected subsets of R, Continuous

functions on connected sets. Local connectedness and arc-wise connectedness. (8 hours)

(RBT Levels: L1, L2 and L3)

Suggested Learning Resources:

Books (Name of the author/Title of the Book/Name of the publisher/Edition and Year)

Text Books

1. P.K. Jain & Khalil Ahamad, “Metric Spaces”. Narosa, 2019.

2. Micheal O; Searcoid, “Metric spaces”. Springer-Verlag, 2009.

Reference Books:

1. Satish Shirali & Harikishan L. Vasudeva, “Metric Spaces”, Springer-Verlag, 2006.

2. E.T. Copson, “Metric spaces”, Cambridge University Press, 1988.

3. P.R. Halmos, “Naive Set Theory”. Springer, 1974.

4. S. Kumaresan, “Topology of Metric spaces”, 2nd edition, Narosa, 2011.

5. G.F. Simmons, “Introduction to Topology and Modern Analysis”. McGraw-Hill, 2004.

Web links and Video Lectures (e-Resources):

• http://nptel.ac.in/courses.php?disciplineID=111

• http://www.class-central.com/subject/math(MOOCs)

• http://academicearth.org/

• VTU e-Shikshana Program

• VTU EDUSAT Program.