## MATH3225 Topology

### 15 creditsClass Size: 101

Module manager: TBA
Email: TBA

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

MATH2020 or MATH2022, or equivalent.

This module is not approved as a discovery module

### Module summary

Topology is the study of those properties of a mathematical space which are unchanged by continuous deformations. Indeed, a topology is the minimal extra structure with which we must equip a set so that the idea of "continuity" makes sense in the first place. In this module we introduce topology in an abstract setting and show how it generalizes the familiar notion of continuity from calculus. We then study various topological invariants of spaces, such as connectedness, path connectedness, compactness and the Hausdorff property. The second half of the module is more algebraic: we describe how a certain group, called the fundamental group, can be associated to each topological space. This assignment has the beautiful property that continuous mappings between spaces naturally induce homomorphisms between their associated groups.

### Objectives

On completion of this module, students should be able to:
- Verify the axioms of a topological space for a range of examples and identify whether a space is Hausdorff, connected, path connected, compact;
- Determine whether a given map between topological spaces is continuous and construct homotopies between simple maps;
- Use topological invariants to prove that suitable pairs of spaces are not homeomorphic;
- Compute the fundamental group of a simple space using Van Kampen's Theorem;
- Determine the induced homomorphism of fundamental groups induced by a suitable continuous map.

### Syllabus

- Review of set notation; indexed collections of sets, their union and intersection;
- Topological spaces: axioms, the usual topology on R, the Hausdorff property, closed sets, continuity, the Glue Lemma;
- New spaces from old: the subspace, product and quotient topologies;
- Topological invariants: connectedness, path connectedness, compactness, Tychonoff's Theorem, the Heine-Borel Theorem for R;
- Homotopy: homotopic maps, homotopy equivalence of spaces, deformation retractions;
- The fundamental group: defintion, the fundamental group of a product, the fundamental group of a circle (statement only)
- The induced homomorphism of a continuous map: definition, contravariance, deformation retractions induce isomorphisms;
- Free products of groups, Van Kampen's Theorem (statement only), computation of fundamental groups of simple spaces (e.g. wedge sums, finite graphs, Polygons with egde identification).

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular exercise sheets.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated