M e t r i c      S p a c e s

OVERVIEW

This page will be used to make announcements and provide copies of handouts, remarks on the textbook, problem sheets and their solutions for this course. Files will be supplied in pdf format.

I welcome feedback in the form of constructive comments or criticism. Just send an email to or talk to me after the lectures.

Information handout

Mathematical questions and questions on this unit (e.g., received by email) are answered in the FAQ-pdf, see Section "Lecture Notes".

Exam papers of the previous years can be found on the Library and Learning Centre Exam Papers Database (type in "MA30041").

FINAL EXAM

The Final Exam took place on 12/1/2009 and can be found here.
Feedback on the Final Exam is available here.

LECTURE NOTES

S. Shirali & H.L. Vasudeva : Metric Spaces
Link to this ebook (via library)

Note: You need to abide by copyright law. Do not download/print the whole book, because if you do our access may be terminated. Each time you need to read a section you can access it via the library catalogue (or directly using SpringerLink).

Here is an overview of topics we have covered so far (version: 1 Dec).

FAQs are answered here (version: 10 Jan).

The powerpoint "polling" presentation used in the revision lecture can be found here.

Here is a handout about the proof of Theorem II.10.

Here is a handout about a nowhere differentiable continuous function obtained using Banach's Contraction Mapping Principle.

Here is a handout about the proof of Theorem VII.5.

Here is a handout about a space-filling curve.

EXERCISE SHEETS

Exercise sheet 10 with solutions (without solutions)

Exercise sheet 9 with solutions (without solutions)

Exercise sheet 8 with solutions (without solutions)

Exercise sheet 7 with solutions (without solutions)

Exercise sheet 6 with solutions (without solutions)

Exercise sheet 5 with solutions (without solutions)

Exercise sheet 4 with solutions (without solutions)

Exercise sheet 3 with solutions (without solutions)

Exercise sheet 2 with solutions (without solutions) -- on the remarks in Question 1

Exercise sheet 1 with solutions (without solutions)

SELF-ASSESSMENT SHEETS

Self-assessment sheet 10

Self-assessment sheet 9

Self-assessment sheet 8

Self-assessment sheet 7

Self-assessment sheet 6

Self-assessment sheet 5

Self-assessment sheet 4

Self-assessment sheet 3

Self-assessment sheet 2

Self-assessment sheet 1

DROP-IN SESSION MATERIAL

Old exam questions 9

Old exam questions 8

Old exam questions 7

Old exam questions 6

Old exam questions 5

Old exam questions 4

Old exam questions 3

Old exam questions 2

Old exam questions 1

ADDITIONAL MATERIALS

Essay-writing competition! Explain metric spaces to everyone!

1st prize:	Christopher H.
2nd prize:	Robert S.

Solve some challenging problems! Have a look at these!

Problem 1	solved by Andrew M. (1 Nov)
Problem 2	solved by Robert S. (3 Nov)
Problem 3	unsolved - "joker" question solved by Andrew M., Bati S. & Robert S.
Problem 4	solved by Anthony M. (3 Nov)
Problem 5	solved by Bati S. (13 Nov)

The programme TeraFractal (for Mac OS X) was used to generate the nice picture in the first lecture.

Wikipedia & MacTutor Links
Maurice René Frechét introduced "metric spaces" in his thesis (1906). [Wikipedia] [MacTutor]
Felix Hausdorff chose the name "metric space" in his influential book from 1914. [Wikipedia] [MacTutor]
Bernardo Bolzano and Augustin Louis Cauchy (in 1817/1821) defined "Cauchy sequences" and "continuity" using ε-δ-notation. [Wikipedia] [MacTutor] & [Wikipedia] [MacTutor]
Richard Dedeking (in 1858/1872, using "Dedekind cuts") and Georg Cantor (in 1872, using Cauchy sequences) showed that ℝ is complete. [Wikipedia] [MacTutor] & [Wikipedia] [MacTutor]
Stefan Banach formulated his Contraction Mapping Principle in 1922. [Wikipedia] [MacTutor]
Karl Weierstraß presented a nowhere differentiable continuous function in 1872. [Wikipedia] [MacTutor]
Giuseppe Peano found a space-filling curve in 1890. [Wikipedia] [MacTutor]
Also check out the BBC Series "The Story of Maths"! [BBC iPlayer]

DIARY

01.12.: equivalence of compactness and sequential compactness in a metric space; (dis)connectedness; intervals in ℝ are connected; path connectedness implies connectedness

24.11.: compactness; nets & total boundedness; sequential compactness; Heine-Borel Theorem; continuous functions on a compact space are uniformly continuous

17.11.: complete subspaces; denseness; contraction; fixed point; Banach's Contraction Mapping Principle; Cantor set; Picard's Theorem

10.11.: Urysohn's Lemma; Lipschitz continuity; homeomorphisms

3.11.: continuous & uniformly continuous functions; characterisation of continuity at a point, continuity & uniform continuity; composition of continuous functions is continuous

27.10.: complement of closed set is open; boundary; sequential characterisation of openness & closedness; topology of subspaces

20.10.: open & closed balls; neighbourhood; open & closed sets; interior points & limit points; interior & closure

13.10.: Cauchy sequences; convergent implies Cauchy implies bounded; complete spaces; ℝⁿ is complete; B(X) and C[a,b] are complete w.r.t. supremum metric; every metric space has a completion (w/o proof)

6.10.: metric subspaces and product spaces; isometry; pseudometric; convergent sequences in metric spaces; equivalent metrics and convergence in product spaces; boundedness and diameter

3.10.: recap session:

M8	ℝn is a real vector space with scalar/inner product and norm & the Cauchy-Schwarz inequality
M7	convergent and Cauchy sequences in ℝ: monotone sequence theorem, Bolzano-Weierstrass, boundedness and convergence of Cauchy sequences
M11	definitions of continuity, uniform continuity and Lipschitz continuity

29.9.: introduction & overview; definition of a metric space; examples