Topology of Euclidean spaces

Description

Subject objectives: The study of the real line and continuous real-valued maps was treated in the course “Introduction to mathematical analysis”. The main purpose of this course is to address the investigation of the topology of Euclidean spaces of any dimension. More specifically: - Studying concepts, methods, and properties of metric spaces, with particular emphasis on the n-dimensional Euclidean space. - Applying techniques related to the convergence of sequences to the study of properties associated with topology. Studying the notion of completeness. - Studying continuous maps between metric spaces, focusing on these between Euclidean spaces. Providing examples of maps that illustrate different properties or serve to define subsets of Euclidean space. Expressing simple geometric transformations analytically. - Studying the concepts of connectedness and compactness. Understand how these concepts allow us to generalize the fact that continuous maps defined on a closed and bounded interval and taking real values reach a maximum and a minimum, as well as all intermediate values between them. Contents: Topic 1. Metric and Euclidean spaces (4 lecture hours) - Metric and topological space. Vector space with inner product. Normed space. Euclidean space. - Cauchy–Schwarz inequality and Minkowski inequality. - Open and closed balls. - Distance between sets. Bounded sets. Diameter. Topic 2. The topology of metric and Euclidean spaces (4 lecture hours) - Open and closed sets. - The topology of metric and Euclidean spaces. - Spaces and subspaces. - Relative topology. Topic 3. Convergence and completeness (4 lecture hours) - Sequences and convergence. Subsequences. - Convergence and topology. - Cauchy sequences. - Completeness. - Completeness of Euclidean spaces. Topic 4. Continuity (8 lecture hours) - Continuity at a point. Global continuity. - Global characterizations of continuity. - Sequential continuity. - Combined maps. - Homeomorphisms. Topological properties. Topic 5. Connectedness (4 lecture hours) - Separation. Connected sets. - Connectedness and continuity. - Path-connected sets. Topic 6. Compactness (4 lecture hours) - Open cover. Compact sets. - Compactness and continuity. - Heine–Borel Theorem.