Materials available from this web page (lecture notes, problem sheets etc.) are copyright © J.F. Feinstein unless otherwise stated.
G13MTS: METRIC AND TOPOLOGICAL SPACES 2004/5
PLEASE NOTE: the current version of G13MTS is at
http://www.maths.nott.ac.uk/MathsModules/G13MTS
Last modified: December 10 2004
 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 realvariable 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.
 Knowledge and understanding

Brief Syllabus:
 Review of the theory of mathematical analysis from earlier modules.
 Introduction to metric spaces
 Introduction to topological spaces
 Subspaces, quotients and products
 Compactness
 Connectedness

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.
 V Bryant, Metric Spaces
 J C Burkill and H Burkill, A Second Course in Mathematical Analysis
 E T Copson, Metric Spaces
 G H Fullerton, Mathematical Analysis
 G. J. O. Jameson, Topology and normed spaces (more advanced)
 B Kripke, Introduction to Analysis
 R G Kuller, Topics in Modern Analysis
 B. Mendelson, Introduction to Topology
 C G C Pitts, Introduction to Metric Spaces
 W Rudin, Principles of Mathematical Analysis
 W A Sutherland, Introduction to Metric and Topological Spaces
Handouts and other documents
 Module information sheet (this document).
 Blowbyblow 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 19978 examination: ps, pdf
 Model answers to the 20034 examination:
ps, pdf
Other handouts/documents will be posted on the module web page as they appear.
Course work
Regular question sheets will be issued. With the exception of the (optional) prize problems, solutions should be handed in to me at the appointed time (shown on each sheet) for marking. The assignments form an essential part of the learning process but do not form part of your formal assessment. The question sheets will be available here as they are issued. Coursework is due in at the end of the Thursday lecture in Teaching Weeks 2, 4, 6, 8 and 10. (Note here that Teaching Week 1 of Autumn Semester is the second week of term.) Prize problems:
ps, pdf
Please note that I did not personally invent these questions. (They are wellknown, 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
Assessment will be by means of a two and a half hour examination in January. There will be five questions, and your best four answers will count. If you answer four of the questions perfectly, then you will obtain full marks. Please note that no calculators will be permitted in the examination. Should a resit examination be required in August/September it will take the same form as above (five questions, best four answers count, no calculators).You may find it useful to look at the exam papers for G13MTS from previous years, but note that the syllabus has changed this year. Nevertheless, the style of question is likely to be similar to those in the exams in the years when Dr Feinstein previously taught G13MTS: 19978, 19989, 19992000 and 20034. Please note that the format of the exam changed to its current form (best four questions from five) between 2000 and 2003
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
 School of Mathematical Sciences Module Information
 Information Gateway, including past exam papers etc.
 Courses Office, examination information
 The University of Nottingham, Library Services (including the online catalogue)
 Blackwell's online bookshop
 Timetable, School of Mathematical Sciences
 University Study Support Centre
 Joel Feinstein's timetable
 Joel Feinstein's recommended mathematics books
 Other useful web pages.
Page maintained by Joel Feinstein, Joel.Feinstein@nottingham.ac.uk, http://explainingmaths.wordpress.com