Undergraduate Programme and Module Handbook 2024-2025

# Module MATH2727: Topology II

## Department: Mathematical Sciences

### MATH2727: Topology II

Type | Open | Level | 2 | Credits | 10 | Availability | Available in 2024/2025 | Module Cap | None. | Location | Durham |
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#### Prerequisites

- Calculus I (Maths Hons) (MATH1081) or Calculus I (MATH1061) AND Linear Algebra I (Maths Hons) (MATH1091) or Linear Algebra I (MATH1071) AND Analysis I (MATH1051)

#### Corequisites

- Complex Analysis II (MATH2011)

#### Excluded Combination of Modules

- None

#### Aims

- To provide an introduction to topology. To build up the theory of topological spaces, or point-set topology, from axioms.
- To improve students' ability to construct proofs.
- To provide background necessary for applying topology in other areas of mathematics.

#### Content

- Topological spaces and continuous functions.
- Open sets, closed sets, limit points and closure, examples of topologies.
- Compact, connected, and Hausdorff spaces.
- The product and quotient topologies.
- Topological groups and group actions.
- Urysohn Metrisation theorem.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module, students will:
- Be able to solve a range of predictable and unpredictable problems in Topology, have an awareness of the abstract concepts of theoretical mathematics in the field of Topology, and have a knowledge and understanding of this subject demonstrated through one or more of the following topic areas:
- Topological spaces.
- Hausdorff, compact, and connected spaces.
- The product and quotient topologies.
- Topological groups and group actions.

Subject-specific Skills:

- In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Spatial awareness.

Key Skills:

- Students will have basic mathematical skills in the following areas: problem solving, abstract reasoning.

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
- Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
- Tutorials provide active engagement and feedback to the learning process.
- Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
- The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 22 | 2 per week in Epiphany and in first week of Easter | 1 hour | 22 | |

Tutorials | 5 | Fortnightly in Epiphany and one in Easter | 1 hour | 5 | ■ |

Problems Classes | 5 | Fortnightly in Epiphany | 1 hour | 5 | |

Preparation and Reading | 68 | ||||

Total | 100 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Examination | 2 hours | 100% |

#### Formative Assessment:

Fortnightly written assignments.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University