ABOUT THE COURSE: The primary objective of this proposed online course is to give the idea of the distance between two elements in an arbitrary set and to extend the concepts, namely, open sets, closed sets, convergence of sequences, compact sets, continuity of functions etc., from real line to a metric space. The course focuses on basic notions of metric spaces and their properties. A large number of both Multiple-Choice Questions and Subjective Questions will be provided for the practice of the students. Several real-world problems will be discussed in the frame work of metric spaces.INTENDED AUDIENCE: B.Sc. / B.C.A. / M.Sc. / M.C.A. / B.TechPREREQUISITES: Basic Real Analysis
Overview
Syllabus
Week 1: Basic Concepts: Definition and examples of metric spaces, bounded and unbounded metric spaces, Distance between sets, Diameter of a set.Week 2:Open and Closed Sets: Open and closed balls, lnterior points and interior of a set, Open set, Neighbourhood of a point, Limit point of a set, Closure of a set, Closed setWeek 3:Boundary of a Set and SubSpaces: Boundary points and boundary of a set, Exterior points and exterior of a set, Subspace of a metric space.Week 4:Convergent Sequences: Sequences and subsequence in a metric space, Convergent and Cauchy sequencesWeek 5:Complete Metric Spaces: Complete metric spaces, Relation between completeness and closedness, Cantor lntersection Theorem, Completion TheoremWeek 6:Dense Sets and Related Results: Dense sets, Separable spaces, Nowhere dense sets, Categories and Baire Category Theorem.Week 7:Compact Metric Spaces: Cover of a metric space, Compact metric spaces, Compact sets and their criterion, Properties of compact setsWeek 8:Finite lntersection Property: Relation between compactness, completeness and closedness, Finite lntersection propertyWeek 9:Some lmportant Theorems: Bolzano-Weierstrass property, Sequential compactness, Totally bounded spacesWeek 10:Continuous Functions: Continuous functions between two metric spaces, Characterizations of Continuous functionsWeek 11:Continuous functions, Uniform continuity: Continuous functions on compact spaces, Uniform continuous functions, Homeomorphism and lsometry,Week 12:Equicontinuity and Fixed Point Theory: Equicontinuity and Ascoli-Arzela Theorem, Fixed points, Banach contraction theorem.
Taught by
Prof. Qamrul Hasan Ansari, Prof. Monirul Islam
