Connectedness in metric spaces, properties of connected sets, connected components. Equivalent forms of compactness of metric spaces. Totally bounded metric spaces, compact spaces. Principle of contraction (Banach's Fixed Point Theorem).
Basic (or Cauchy) sequences, complete metric spaces. The topology of a metric space, the concept of a topological space. Open balls, closed balls, interior, closed hull and boundary, accumulation points and derived set. Functions between metric spaces, continuous functions, characterization of continuity via sequences, uniform continuity of functions. Sequences in metric spaces, subsequences, convergence of sequences. Metrics in vector spaces induced by norms. Metric spaces, definition, examples, basic properties. Promoting analytical and synthetic thinking.Promoting creative and inductive thinking.Working in an interdisciplinary environment.Analysis and synthesis of data and information.These concepts, as well as the proofs of the related results, are given in such a way that, wherever possible, they can be easily and without major changes adapted toTopological Spaces. Topics which are covered are convergence, continuity, completeness, total boundness, compactness, separability and connectedness. An in-depth study of Metric spaces, in addition to preparing the student to accept the abstract structures of General Topology, helps him to better understand the structure of the Euclidean space n, which is studied the same time in the Multi-Variable Infinitesimal Calculus. The didactic approach here is to initially give the theory of metric spaces and then, as a mere reference, an introduction to General Topology. In the last few years, Topology has been increasingly used in the creation of mathematical models that serve research applied disciplines such as Economics, Meteorology, Insurance Mathematics, Epidemiology in Medicine, etc. Topology is a powerful tool for research and expression in all branches of Mathematical Science. See eCourse, the Learning Management System maintained by the University of Ioannina.
Is the Course Offered to Erasmus Students
Lectures (Weekly Teaching Hours: 5, Credits: 7.5) 4 Teaching and Learning Methods - Evaluation.
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