Department of Mathematics MATH319-11A

Topics in Pure Mathematics

Class Web Site

Today Friday will be a topology tutorial in G3.22. Come and get me round 4pm if you want to participate - I will be in the Tea room on level 2. Some extra notes were distributed covering the last topic, namely that all metrizable topologies are normal. Assignment 2s has been corrected and the solutions are linked below.

In the final examination you are to choose any 6 out of the 8: 6 questions on number theory and 1 each on generating functions and topology. The list for the number theory questions will be the same as for math509 - see the web page later this week. The generating function question will be somewhat like parts of 1 and 2 in Assignment 1s. The topology question will involve all definitions and theorem statements for what we have covered and proofs for parts of the following list:

  1. The open balls in a metric space are a base for a topology,
  2. Compact Hausdorff spaces are normal,
  3. In a compact space closed subsets are compact,
  4. If X is a toplogical space then the sum of two continuous functions to R is continuous,
  5. If a metric space is sequentially compact it is totally bounded,
  6. Every metric space is Hausdorff and normal in it induced topology.
I will be available on Tuesday 7th June next week and on Friday morning 10th June for test/exam preparation assistance.

We will not cover all of the material on topology linked below.

The paper outline is linked here.


Professor Kevin Broughan, Office G3.22, Tel 838-4423, email

Office Hours: Thursday 3.10-5pm during teaching weeks in G3.22.

Lecture Times:

  1. Monday 11.00 a.m. - 11.50 a.m. (number theory with MATH509)
  2. Tuesday 5.10 p.m. - 6.00 p.m. (number theory with MATH509)
  3. Thursday 5.10 p.m. - 6.00 p.m.(normally a number theory tutorial with MATH509)
  4. Friday 4.10pm-5.00 p.m. (MATH319 special lectures)

Special lectures topics

Lecture Notes for the Friday lectures are linked here.

  1. Enumeration,
  2. Generating Functions I,
  3. Generating Functions II,
  4. Stirling and Bell numbers, Polya enumeration, sieving.
  5. Polya enumeration example, inclusion/exclusion example.
  6. Topology I: page 4, 5, 6, 7, 8, 9.
  7. Metric topologies, closure, interior and boundary: page 10, 11, 12, 13, 14, 15. 16.
  8. Continuity in topological and metric spaces: page 17, 18, 19, 20, 21, 22. 23.
  9. Compactness and the Heine-Borell theorem: page 24, 25, 26, 27, 28, 29.
  10. Connectness: page 37, 38, 39, 40, 41, 42, 43.
  11. Separable and second countable spaces: page 1, 2, 3, 4, 5.
  12. Ursohn's metrization theorem: 6, 7.
  13. Contraction mappings and applications: 8, 9, 10.

Information sources

A book by Herbert Wilf called Generatingfunctionology is linked here. For topology texts see the list for the paper Metric Spaces on my home page.

Assessment for 2011:

The internal assessment will consist of six assignments each worth 8% and a test worth 52%. Assignments are due by or before 6pm on the specified dates. The final examination, scheduled for a date in June, will be weighted 1:1 or 1:0, which ever is more favourable for the student.

Dates and values of the Assignments and the internal test:

  1. Assignment 1: Out Thursday 10th March, back Thursday 17th March, value 8%,
  2. Assignment 2: Out Thursday 24th March, back Thursday 31st March, value 8%,
  3. Assignment 1s: Out Friday 1st April, back Friday 8th April, value 8%
  4. Recess
  5. Test 1: Thursday 5th May in G3.33 at 5pm, value 52%.
  6. Assignment 3: Out Thursday 12th May, back Thursday 19th May, value 8%,
  7. Assignment 2s: Out Friday 20th May, back Friday 27th May, value 8%,
  8. Assignment 4: Out Tuesday 26th May, back Thursday 2nd June, value 8%,
  9. Final Examination: in June after the study break.

Assignments for 2011:

  1. 1s: First assignment questions, solutions,
  2. 2s: Second assignment questions, solutions.

Assignments from 2010:

  1. 1s: First assignment questions and solutions,
  2. 2s: Second assignment questions, solutions.

Kevin Broughan

3rd June 2011