# GENERAL TOPOLOGY - 2024/5

Module code: MATM042

## Module Overview

This module gives a self-contained, rigorous, formal treatment of basic topics in point set topology. Topology is an important topic in modern mathematics and the module will give the thorough grounding in this field. The module will expose students to abstract, general mathematical arguments and techniques.

This module builds on material from MAT2004 Real Analysis 2. It also complements the module material in MAT3009 Manifolds and Topology.

### Module provider

Mathematics & Physics

PRINSLOO Andrea (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 64

Lecture Hours: 33

Tutorial Hours: 5

Guided Learning: 15

Captured Content: 33

Semester 2

None.

## Module content

Indicative content includes:

• Open sets and neighbourhoods: Topological spaces and open sets. Metric spaces. Neighbourhoods and neighbourhood bases of points. Bases of topologies. First and second countable topological spaces. Closed sets. Interiors, closures and boundaries of subsets of topological spaces. Dense subsets.

• Continuous maps, connected spaces and separation properties: Continuous maps. Convergence. Sequences and nets. Connected topological spaces. Separation properties and Hausdorff spaces.

• Compactness: Covers. Compact topological spaces and compact subsets. Heine-Borel Theorem. Finite intersection property.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 min) 20
Examination End-of-Semester Examination (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of subject knowledge, and recall of key definitions and theorems of point set topology.

• The ability to construct simple proofs similar to those in the module.

• The ability to identify and use the appropriate methods to solve problems relating to topology.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcome 1, 4 and 5.

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 5.

Formative assessment

There are two formative unassessed courseworks over an eleven week period, designed to consolidate student learning.

Feedback

Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in tutorials and office hours.

## Module aims

• Introduce students to a rigorous and formal treatment of point set topology.
• Develop students' understanding of rigorous proofs in the context of topology.

## Learning outcomes

 Attributes Developed 001 Students will demonstrate an understanding of topological spaces and open sets, neighbourhoods and bases, closed sets, interiors, closures and boundaries of subsets of topological spaces. KC 002 Students will demonstrate an understanding of continuous maps, connected spaces and separation properties of topological spaces. KC 003 Students will demonstrate an understanding of compact topological spaces and compact subsets. KC 004 Students will be able to solve examples and problems related to point set topology. KC 005 Students will be able to construct simple proofs similar to those encountered in the module. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Introduce students to the fundamental topics in point set topology.

• Provide students with experience of methods used to interpret, understand and solve problems in topology.

The learning and teaching methods include:

• Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Five biweekly one-hour tutorials per semester. These tutorials provide an opportunity for students to gain feedback and assistance with the exercises which complement the module notes.

• Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these courseworks as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MATM042

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM042 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: The module MATM042 equips students with skills which significantly enhance their employability. The mathematical proficiency gained will hone their critical thinking and problem-solving abilities. Students will learn to evaluate complex topological problems, break them into manageable components, and apply the theory of abstract point set topology and logical reasoning to arrive at solutions. These are highly sought after skills in many professions.

Global and Cultural Capabilities: Student enrolled in MATM042 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MATM042 is a module which demands a rigorous approach to point set topology, to which students will learn to adapt. They will gain skills in formulating rigorous mathematical proofs and analysing abstract topological problems using point set topology and lateral thinking. Students will complete assessments which challenge them and build resilience.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MSc 2 Optional A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.