G13MTS: METRIC AND TOPOLOGICAL SPACES 2004/5

• Module Information
• Handouts and other documents
• Coursework
• Assessment
• Miscellaneous links of interest

Module information for 2004/2005

• Credits: 15
• Duration: approximately 33 lectures (including at least one problem class/question and answer session), three lectures a week in Autumn Semester, starting Monday 27/9/04
• Lecturer: Dr J. F. Feinstein, room C301 Maths/Physics, email Joel.Feinstein@nottingham.ac.uk
• Lecture times: Monday 17:00 M&P Building Room B23, Tuesday 9:00 Pope Building Room A1 and Thursday 12:00 noon M&P Building Room B23.
• Office hours: Dr J.F. Feinstein, room C301 M&P: See the web page http://www.maths.nott.ac.uk/personal/jff/ttnow.html for details of Dr Feinstein's timetable and office hours.
• Module noticeboard: The module noticeboard is available on the Student Portal Module Page for G13MTS. To find this, log in to http://my.nottingham.ac.uk, choose the My Teaching tab, and click on the View your modules button.
• Brief content description: Metric space generalises the concept of distance familiar from Euclidean space. It provides a notion of continuity for functions between quite general spaces. This allows us to consider fundamental concepts such as completeness, compactness and connectedness, and to prove key results concerning them. These in turn throw new light on staples of real-variable theory such as the Intermediate Value Theorem. With topological spaces we are even able to remove the reliance on distance, placing the above ideas in a context which is much more general still. Perhaps surprisingly, this very abstract setting renders some of the above ideas easier to handle!
• Prerequisites: G1BMAN: Mathematical Analysis
• Module Aims:This module is part of the Analysis Pathway within Pure Mathematics, providing an introduction to some of the principal definitions and theorems concerning metric and topological spaces. It is a key prerequisite to further modules within the Analysis pathway (G13HIL, G14FUN, G14TOP and G14NCA).
• Learning Outcomes:
  A student who completes this module successfully should be able to:
  • Knowledge and understanding
    • define and state some of the basic definitions of concepts concerning metric and topological spaces such as compactness, connectedness, completeness;
    • state some of the principal theorems as treated in the module; use some of the basic definitions and principal theorems in the investigation of examples;
    • prove basic propositions concerning those aspects of metric and topological spaces treated in the module.
  • Intellectual skills
    • apply complex ideas to familiar and to novel situations;
    • work with abstract concepts and in a context of generality;
    • reason logically and work analytically;
    • perform with high levels of accuracy;
    • transfer expertise between different topics in mathematics.
  • Professional skills
    • select and apply appropriate methods and techniques to solve problems;
    • justify conclusions using mathematical arguments with appropriate rigour;
    • communicate results using appropriate styles, conventions and terminology.
  • Transferable skills
    • communicate with clarity;
    • work effectively, independently and under direction;
    • analyse and solve complex problems accurately;
    • adopt effective strategies for study.
  
  
• Brief Syllabus:
  1. Review of the theory of mathematical analysis from earlier modules.
  2. Introduction to metric spaces
  3. Introduction to topological spaces
  4. Subspaces, quotients and products
  5. Compactness
  6. Connectedness
  7. Complete metric spaces
     
  
  
• Books: Of the following, the books by Mendelson and Sutherland are the most appropriate: Sutherland's book is highly recommended. Both of these books should be available in the library, and Sutherland will be in the bookshop. The other books are in the library and are all worth a look.
  
  1. V Bryant, Metric Spaces
  2. J C Burkill and H Burkill, A Second Course in Mathematical Analysis
  3. E T Copson, Metric Spaces
  4. G H Fullerton, Mathematical Analysis
  5. G. J. O. Jameson, Topology and normed spaces (more advanced)
  6. B Kripke, Introduction to Analysis
  7. R G Kuller, Topics in Modern Analysis
  8. B. Mendelson, Introduction to Topology
  9. C G C Pitts, Introduction to Metric Spaces
  10. W Rudin, Principles of Mathematical Analysis
  11. W A Sutherland, Introduction to Metric and Topological Spaces

Handouts and other documents

• Module information sheet  (this document).
• Blow-by-blow account of the module (now complete): ps, pdf
• Frequently Asked Questions (FAQ), 4 questions and answers so far, last question and answer added November 30 2004: ps, pdf
• Lecture notes (now complete, 63 pages): ps, pdf
  These notes will be updated as the module progresses. They will include most definitions and statements of results and examples, but for full details you will need to attend lectures and/or find the details in textbooks.
• Module Slides (Now complete, 183 slides, 4 slides per page): ps, pdf
• Uniform continuity for functions between subsets of the real line: ps, pdf
• Proof of Lemma 5.5: ps, pdf
• Model answers to the 1997-8 examination: ps, pdf
• Model answers to the 2003-4 examination: ps, pdf
  Other handouts/documents will be posted on the module web page as they appear.
  

Course work

• Prize problems: ps, pdf
  Please note that I did not personally invent these questions. (They are well-known, and mostly rather hard.)
  Prize problem 4 has been correctly solved by Thomas Nolden
  Prize problem 11 has been correctly solved by Thomas Nolden
  
• Question Sheet 1: ps, pdf
• Question Sheet 2: ps, pdf
• Question Sheet 3: ps, pdf
• Question Sheet 4: ps, pdf
• Question Sheet 5: ps, pdf

G13MTS Assessment

Miscellaneous links which may be of interest:

• G12RAN: Real Analysis (a module on real analysis which no longer exists, but which used to be taught by Dr Feinstein)
• Catalogue of Modules, entry for G13MTS: Metric and Topological Spaces [an error occurred while processing this directive]

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