Metric Spaces and their Topology (MAY413)
General
School |
School of Science |
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Academic Unit |
Department of Mathematics |
Level of Studies |
Undergraduate |
Course Code |
MAY413 |
Semester | 4 |
Course Title |
Introduction to Topology |
Independent Teaching Activities |
Lectures (Weekly Teaching Hours: 5, Credits: 7.5) |
Course Type |
General Background |
Prerequisite Courses | - |
Language of Instruction and Examinations |
Greek |
Is the Course Offered to Erasmus Students |
Yes (in English) |
Course Website (URL) |
http://www.math.uoi.gr/GR/studies/undergraduate/courses/413.html |
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.
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General Competences |
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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 | ||||||||||
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Use of Information and Communications Technology |
Use of ICT for presentation of essays and assignments | ||||||||||
Teaching Methods |
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Student Performance Evaluation |
Written examination at the end of the semester including theory and problems-exercises. |
Attached Bibliography
- 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.
- X. Καρυοφύλη και Χ. Κωνσταντιλάκη, Τοπολογία, Ι, ΙΙ, Εκδόσεις Ζήτη, Θεσσαλονίκη 1990.
- 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.
- Π. Τσαμάτος, Τοπολογία, 2η ΄Εκδοση, Παναγιώτης, Εκδόσεις Τζιόλα.