Metric spaces, open balls and open sets, limit and cluster points, closed sets, dense sets, complete metric spaces, completion of a metric space, Continuity, uniform continuity, Banach contraction principle, Compactness, Connectedness, pathconnected sets. Sequences of functions, Pointwise convergence and uniform convergence, Arzela-Ascoli Theorem, Weierstrass Approximation Theorem, power series, radius of convergence, uniform convergence and Riemann integration, uniform convergence and differentiation, Stone Weierstrass theorem for compact metric spaces.
- G. F. Simmons, “Introduction to Topology and Modern Analysis”, Tata McGraw-Hill, 2013.
- S. Kumaresan, “Topology of Metric Spaces”, Narosa Publishing House, 2005.
- R. R. Goldberg, “Methods of Real Analysis”, John Wiley & Sons, 1976.
- G. B. Folland, “Real Analysis”, Wiley-Interscience Publication, John Wiley & Sons, 1999.