# Metric Spaces and their Topology (MAY413)

### General

School School of Science Department of Mathematics Undergraduate MAY413 4 Metric Spaces and their Topology Lectures (Weekly Teaching Hours: 5, Credits: 7.5) General Background - Greek Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes Topology is a powerful tool for research and expression in all branches of Mathematical Science. 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. The didactic approach here is to initially give the theory of metric spaces and then, as a mere reference, an introduction to General Topology. 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. Topics which are covered are convergence, continuity, completeness, total boundness, compactness, separability and connectedness. 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. Analysis and synthesis of data and information Autonomous work Teamwork Working in an interdisciplinary environment Promoting creative and inductive thinking Promoting analytical and synthetic thinking Production of new research ideas.

### Syllabus

Metric spaces, definition, examples, basic properties. Metrics in vector spaces induced by norms. Diameter of a set, distance of sets. Sequences in metric spaces, subsequences, convergence of sequences. Functions between metric spaces, continuous functions, characterization of continuity via sequences, uniform continuity of functions. Open balls, closed balls, interior, closed hull and boundary, accumulation points and derived set. The topology of a metric space, the concept of a topological space. Basic (or Cauchy) sequences, complete metric spaces. Principle of contraction (Banach's Fixed Point Theorem). Totally bounded metric spaces, compact spaces. Equivalent forms of compactness of metric spaces. Properties of compact spaces. Separable metric spaces. Connectedness in metric spaces, properties of connected sets, connected components.

### Teaching and Learning Methods - Evaluation

Delivery

Face-to-face

Use of Information and Communications Technology

Use of ICT for presentation of essays and assignments

Teaching Methods
Lectures 65
Solving exercises at home 22.5
Individual study 100
Course total 187.5
Student Performance Evaluation

Written examination at the end of the semester including theory and problems-exercises.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• K. W. Anderson and D. W. Hall, Sers, Sequences and Mappings, John Wiley and Sons, Inc. New York 1963.
• V. Arkhangel’skii and V.I. Ponomarev, Fundamentals of General topology, D. Reidel Publishing Company, 1983.
• G. Buskes and A. van Rooij, Topological Spaces, Springer-Verlag, New York, 1197.
• D. C. J. Burgess, Analytical Topology, D. Van Nostrand Co. Ltd., London, 1966.
• N. L. Carothers, Real Analysis, Cambridge University Press, 2000.
• E. Copson, Metric Spaces, Cambridge University Press, 1968.
• J. Diedonne, Foundations of Modern Analysis, Academic Press, New York, 1966.
• J. Dugudji, Topology, Allyn and Bacon Inc., Boston, 1978.
• W. Franz, General Topology, G. Harrap and Co. Ltd. London 1965.
• J. R. Giles, Introduction to the Analysis of Metric Spaces, Cambridge University Press, 1989.
• S.-T. Hu, Introduction to General Topology, Holden-Day Inc. San Francisco, 1966.
• T. Husain, Topology and Maps, Plenum Press, New York, 1977.
• K. D. Joshi, Introduction to General Topology, Wiley Eastern Limited, New Delhi, 1986.
• Ι. Kaplansky, Set Theory and Metric Spaces, Allyn and Bacon Inc., Boston, 1975.
• R. L. Kasriel, Undergraduate Topology, W. B. Saunders Co. Philadelphia, 1971.
• J. L. Kelley, General Topology, D. Van Nostrand Co. Inc., Toronto 1965.
• S. Lipschutz, Theory and Problems of General Topology, Schaum’s Outline Series, New York, 1965.
• Mwndelson, Introduction to Topology, Prentice-Hall Inc. New Jersey, 1975.
• M. G. Murdeshuar, General Topology, Wiley Eastern Limited, New Delhi, 1986.
• M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge University Press, 1964.
• Α. W. Schurle, Topics in Topology, North Holland, New York, 1979.
• Β. Στάϊκος, Μαθήματα Μαθηματικής Αναλύσεως Μέρος Ι και Μέρος ΙΙ, Ιωάννινα, 1981.