# MA3F1 Introduction to Topology

**Lecturer: **Colin Sparrow

**Term(s):** Term 1

**Status for Mathematics students:** List A

**Commitment:** 30 one-hour lectures

**Assessment:** 85% 3 hour examination, 15% assignments

**Formal registration prerequisites: **None

**Assumed knowledge: **

MA260 Norms, Metrics and Topologies or MA222 Metric Spaces:

- Topological spaces
- Continuous functions
- Homeomorphisms
- Compactness
- Connectedness

MA136 Introduction to Abstract Algebra:

- Groups
- Subgroups
- Homomorphisms and Isomorphisms

**Useful knowledge: **

- Interest in geometry e.g., MA243 Geometry
- More experience with groups e.g., MA251 Algebra I: Advanced Linear Algebra or MA249 Algebra II: Groups and Rings

**Synergies: **The following modules go well together with Introduction to Topology:

**Leads to: **The following modules have this module listed as **assumed knowledge** or **useful background:**

- MA3H6 Algebraic Topology
- MA475 Riemann Surfaces
- MA4H4 Geometric Group Theory
- MA4J7 Cohomology and Poincare Duality

**Content**: Topology is the study of properties of spaces invariant under continuous deformation. For this reason it is often called "rubber-sheet geometry''. The module covers: topological spaces and basic examples, compactness, connectedness and path-connectedness, identification topology, Cartesian products, homotopy and the fundamental group, winding numbers and applications, an outline of the classification of surfaces.

**Aims**: To introduce and illustrate the main ideas and problems of topology.

**Objectives**:

- To explain how to distinguish spaces by means of simple topological invariants (compactness, connectedness and the fundamental group)
- To explain how to construct spaces by gluing and to prove that in certain cases that the result is homeomorphic to a standard space
- To construct simple examples of spaces with given properties (e.g. compact but not connected or connected but not path connected).

**Books**:

Chapter 1 of Allen Hatcher's book* Algebraic Topology*

For more reading, see the Moodle Pages (link below). MA Armstrong, *Basic Topology* Springer (recommended but not essential).