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)

Teaching-Learning Process: Chalk and talk method / PowerPoint Presentation

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)

Teaching-Learning Process: Chalk and talk method / PowerPoint Presentation

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)

Teaching-Learning Process: Chalk and talk method / PowerPoint Presentation

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)

Teaching-Learning Process: Chalk and talk method / PowerPoint Presentation

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)

Teaching-Learning Process: Chalk and talk method / PowerPoint Presentation