Upper bounds, maximum elements, least upper bound-the completeness axiom-some properties of the supremum-properties of the integers deduced form the completeness axiom-the Archimedean property of the real number system-Absolute values and the triangle inequality- the Cauchy Schwarz Inequality. Some basic Notions of set theory: Countable and uncountable sets -Uncountability of the real number system-Set Algebra – Countable collections of countable sets Elements of Point set Topology: Introduction, Euclidean space Rn – open balls and open sets in Rn-closed sets and adherent points – The Bolzano – Weierstrass theorem.

Lindelof covering theorem – the Heine Borel covering Theorem – Compactness in Rn – Metric Spaces –point set topology in metric spaces – compact subsets of a metric space –Boundary of a set.

Introduction-Convergent sequences in metric space – Cauchy sequences –Completeness sequences – complete metric Spaces-Limit of a function – Continuous functions–continuity of composite functions. Continuous complex valued and vector valued functions.

Examples of continuous functions - Continuity and inverse images of open or closed sets – functions continuous on compact sets –Topological mappings – Bolzano’s Theorem

Definition of derivative – Derivative and continuity – Algebra of Derivatives –the chain rule-one sided derivatives and infinite derivatives-functions with nonzero derivative –zero derivatives and local extrema- Roll’s theorem –The mean value theorem for derivatives -Taylor’s formula with remainder.