Basic Topology : Finite, countable and uncountable sets. Metric spaces, compact sets. Perfect sets. Connected sets.
Sequences and Series : Convergent sequences (in metric spaces). Subsequences. Cauchy sequences. Upper and lower limits of a sequence of real numbers. Riemann’s Theorem on Rearrangements of series of real and complex numbers.
Continuity : Limits of functions (in metric spaces). Continuous functions. Continuity and compactness. Continuity and connectedness. Monotonic functions.
The Riemann-Stieljes Integral : Definition and existence of the Riemann-Stieltjes integral. Properties of the integral. Integration of vector-valued functions. Rectifiable curves.
Sequences and Series of Functions : Problem of interchange of limit processes for sequences of functions. Uniform convergence. Uniform convergence and continuity. Uniform convergence and integration. Uniform convergence and differentiation. Equicontinuous families of functions, The Stone-Weierstrass theorem.