The course provides advanced knowledge of concepts and theorems in advanced analysis. The concept of topology is introduced in metric spaces. The concepts of compactness and continuity are essential. Thereafter real-valued functions defined on metric spaces are studied, with a focus on continuity and function sequences. Central theorems are Heine-Borel covering theorem, Urysohn's lemma and Weierstrass' approximation theorem. The concept of differentiability of vector-valued functions is introduced and the inverse and implicit function theorems are proved.