Module manager: Joao Faria Martins
Email: j.fariamartins@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2026/27
MATH3225:Topology.
This module is not approved as an Elective
Topology studies shapes, the ways we can map between them continuously, and which properties are preserved under their “homeomorphisms” (isomorphisms in topology). E.g. a rugby ball and a ball despite not having exactly the same shape are homeomorphic, but a doughnut and a sphere are fundamentally different. Spaces exist, however, that cannot be embedded in Euclidean space, and their points can be abstract objects, like functions. A topological space is a set with just enough structure to define continuity. The concept is extremely flexible and far-reaching. In this module we also cover topological properties of spaces, for example connectedness and compactness. At the end of the module, we study algebraic topology, and solve topological problems by moving them to an algebraic setting. We elucidate some applications, e.g. we cannot continuously retract a disk onto its boundary; hence a drum cannot collapse to its hoop.
The objective of this module is to give a precise definition of topological spaces, illustrated with several examples and applications, ranging from classical analysis and geometric topology in Euclidean space to functional analysis, and logic and set theory.
It is expected that the focus of the module may significantly vary from year to year, depending on the lecturer’s expertise and student’s interests. Core topics will likely include classical point-set topology and the study of topological properties, such as connectedness, compactness, and the Hausdorff property.
It is also expected that students will meet some of the functors arising in algebraic topology, such as the fundamental group, and learn how they can be used to solve hard topological problems by moving them to a more algebraic and discrete setting.
Subject specific learning outcomes:
On successful completion of the module students will have demonstrated the following learning outcomes relevant to the subject:
To develop the skills of rigorous logical argument and problem-
solving in the context of point-set topology
Identify whether a family of subsets of a set is a topology,
Prove elementary results about topologies and subsets of a topological space (and their closure, interior, limit points, etc),
Evaluate whether a function is continuous, including in comparison with the definition of continuity of real-valued functions.
Determine when a topological space is connected, path-connected, compact, separable, Hausdorff, and first-countable.
Prove results concerning quotient, product and subspace topologies.
Construct homotopies between simple maps.
Determine the homomorphism of fundamental groups induced by a continuous map.
Judge whether two spaces can be homeomorphic by using topological invariants.
Compute whether two maps can be homotopic by means of their induced map on fundamental groups.
Skills learning outcomes:
On successful completion of the module students will have demonstrated the following skills:
a) Construct clear mathematical arguments, identifying conclusions and assumptions.
b) Understand that an unambiguous language is paramount in mathematics.
c) Appreciate the power of abstraction and generalisation in problem-solving (enabling the same technique to be transferred across a multitude of structures).
d) By writing down proofs in an engaging and clear way, this module will also improve students' communication skills when presenting arguments in a clear and precise manner.
Review of set-theoretical notions; union and intersections of arbitrary families of subsets.
Definition of topological spaces, examples including the usual topology in the real line and plane, metric topologies and non-metric topologies.
New topological spaces from old: product, subspace and quotient topology, examples.
Separable spaces, first-countable spaces, and Hausdorff spaces.
Connectedness, path-connectedness, and compactness, examples including the product of two topological spaces.
Compactness in the real line: Lebesgue numbers, Heine-Borel property.
Homotopy between paths; fundamental group of a topological space, dependence on base-points.
Deformation retractions.
Examples including the fundamental group of a circle, of a punctured plane, and of a sphere.
Homomorphisms of fundamental groups induced by continuous maps. Applications.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
117
The students will submit exercise series every two weeks. The exercise series will be marked for feedback and general feedback will be provided in problem classes. In particular, one lecture every two weeks will be a problem class.
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
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