# MA260 Norms, Metrics and Topologies

**Lecturer: **Joel Moreira

**Term(s): **Term 2

**Status for Mathematics students: **Core

**Commitment: **30 lectures

**Assessment: **85% by 2-hour June examination, 15% coursework

**Formal registration prerequisites: **None

**Assumed knowledge: **

MA141 Analysis 1 and MA139 Analysis 2:

- Sequences
- Convergence
- Cauchy sequences
- Series
- Continuous functions
- Differentiable functions

- Set theory
- Proofs
- Cardinality

- Pointwise and uniform convergence of sequences of functions
- Open and closed sets in ${\mathbb R}^n$

**Synergies: **The following module goes well together with Norms, Metrics & Topologies:

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

- MA254 Theory of ODEs
- MA3D9 Geometry of Curves and Surfaces
- MA3B8 Complex Analysis
- MA3G6 Commutative Algebra
- MA3G7 Functional Analysis I
- MA3G8 Functional Analysis II
- MA3H6 Algebraic Topology
- MA3D4 Fractal Geometry
- MA359 Measure Theory
- MA3J2 Combinatorics II
- MA3H5 Manifolds
- MA3G1 Theory of Partial Differential Equations
- MA3K1 Mathematics of Machine Learning
- MA3F1 Introduction to Topology
- MA3H3 Set Theory
- MA448 Hyperbolic Geometry
- MA4E0 Lie Groups
- MA4C0 Differential Geometry
- MA427 Ergodic Theory
- MA4M3 Local Fields
- MA4H4 Geometric Group Theory
- MA424 Dynamical Systems
- MA4M7 Complex Dynamics

**Content: **To introduce the notions of Normed Space, Metric Space and Topological Space, and the fundamental properties of Compactness, Connectedness and Completeness that they may possess. Students will gain knowledge of definitions, theorems and calculations in

- Normed, Metric and Topological spaces
- Open and closed sets and their relation to continuity
- Notions of Compactness and relations to continuous maps
- Notions of Connectedness and relations to continuous maps
- Notions of Completeness and relations to previous topics in the module.

The module comprises the following chapters:

- Normed Spaces
- Metric Spaces
- Open and closed sets
- Continuity
- Topological spaces
- Compactness
- Connectedness
- Completeness

**Learning Outcomes:**

- Demonstrate understanding of the basic concepts, theorems and calculations of Normed, Metric and Topological Spaces.
- Demonstrate understanding of the open-set definition of continuity and its relation to previous notions of continuity, and applications to open or closed sets.
- Demonstrate understanding of the basic concepts, theorems and calculations of the concepts of Compactness, Connectedness and Completeness (CCC).
- Demonstrate understanding of the connections that arise between CCC, their relations under continuous maps, and simple applications.

**Books:**

1. W A Sutherland, *Introduction to Metric and Topological Spaces*, OUP.

2. B. Mendelson (1991) *Introduction to Topology. *Dover.