# Topology

Course Code : | 1003WETATP |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 2nd semester |

Sequentiality: | |

Contact hours: | 30 |

Credits: | 3 |

Study load (hours): | 84 |

Contract restrictions: | No contract restriction |

Language of instruction: | Dutch |

Exam period: | exam in the 2nd semester |

Lecturer(s) | Werner Peeters |

### 1. Prerequisites *

At the start of this course the student should have acquired the following competences:

an active knowledge of

specific prerequisites for this course

an active knowledge of

- Dutch

- English

- general knowledge of the use of a PC and the Internet

specific prerequisites for this course

The following course contents precede this course:

- The knowledge about set theory from Discrete wiskunde
- The notions about metric and normed spaces and continuity from Multivariate calculus

### 2. Learning outcomes *

- The students know a large selection of relevant definitions from general topology, as well as the implications between them
- The students can reproduce the proofs of all implications and equivalenes, and highlight their importance.
- The students know a large selection of relevant examples and counterexamples, and can test the topological properties of a given topology.

### 3. Course contents *

The first part of the course consists of:

- an enumeration of the five equivalent descriptions of a topology (with open and closed sets, neighbourhoods, the interior and closure operator),
- a study of the notions of convergence and adherence in a general topological space by means of filters and ultrafilters,
- the equivalent descriptions of continuity in a topological space,
- initial and final structures, with peculiar attention to products and subspaces on the one hand, and quotient spaces and (to a lesser extent) disjoint sums on the other.

In the second part, an overview will be given of the topological properties, i.e. poperties that are preserved onder homomorphisms, and an extensive classification will be made. We will distinguish:

- countability properties: A2, A1 and separability
- separation axioms: T0 (Kolmogorov), T1 (Fréchet), T2 (Hausdorff), regularity, T3, complete regularity, T3 1/2, normality, T4 and the Urysohn lemma, as well as some important corollaries such as the closed graph theorem
- compactness properties: compactness, sequential compactness, local compactness the Alexandroff-compactification, Lindelöf, sigma-compact, Baire spaces and countable compactness
- Connectedness properties

### 4 International dimension*

### 5. Teaching method and planned learning activities

Class contact teachingLectures

### 6. Assessment method and criteria

ExaminationOral with written preparation Closed book

### 7. Study material *

#### 7.1 Required reading

W. Peeters. Algemene Topologie

**7.2 Optional reading**

The following study material can be studied voluntarily :J.R. Munkres. Topology, 2nd edition. Prentice Hall, 2000

G. Preuss. Allgemeine Topologie. Hochschultext. Springer Verlag Berlin, Heidelberg, New York, 1972

S. Willard. General Topology, revised edition. Dover Publications, 1998

### 8. Contact information *

dr Werner Peeters

Dept. Wiskunde en Informatica

Campus Middelheim gebouw G lokaal G3.14

Tel. 03/265.32.93

E-mail: werner.peeters@uantwerpen.be