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G13MTS: METRIC AND TOPOLOGICAL SPACES 2004/5
URL:
http://www.maths.nott.ac.uk/personal/jff/G13MTS/index.html
Last modified: December 10 2004
Many of the documents for this module will be available on the web
in both ps (postscript) and pdf format (portable document format).
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.

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:
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