## Description

**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.