Course Code : | 9001VUBCTO |

Study domain: | Mathematics |

Location: | Not at UA, but at Vrije Universiteit Brussel |

Academic year: | 2019-2020 |

Semester: | 2nd semester |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | Exam contract not possible |

Language of instruction: | Dutch |

Exam period: | exam in the 1st and/or 2nd semester |

Lecturer(s) | Eva Colebunders Bob Lowen |

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

specific prerequisites for this course

specific prerequisites for this course

The student needs to have succeeded in Advanced Topology and in the main fundamental courses of the Bachelor of Science in Mathematics.

- The student can give an interpretation of basic concepts from category theory such as, special objects and morphisms, (co)limits or factorization systems to constructs, i.e. concrete categories over sets, and can apply these concepts in known situations of (Hausdorff) topological or uniform spaces.
- The student has insight in the meaning of the notion topological construct, i.e. in the existence of unique initial lifts for structured sources and knows examples within topology or also in other domains of mathematics. The student recognizes the differences with categories from algebra.
- The student knows the meaning of the property that the embeddings functor from a full and isomorphism closed subconstruct is adjoint or coadjoint and knows characterizations of (extremal) epireflective, concretely reflective or concretely coreflective subconstructs.
- The student has insight in the notion cartesian closedness for topological constructs and knows examples, in particular of subconstructs or superconstructs of Top that are cartesian closed.
- The student has an overall insight in the material, has a deep understanding of new concepts and results and is aware of the connection between de various concepts.
- The student has insight in the relation to analogous concepts as they were introduced in other courses and is able to independently investigate the relation with known concrete situations in the context of topological or uniform spaces.
- The student can complete proofs that are left as an exercise or that are only partially explained in the syllabus or in class. The missing arguments can be filled in independently.
- The student can independently solve problems: he/she is able to recognize a problem, to choose an appropriate strategy, to select the most suitable method.

Subjects will be chosen from the following list:

(1) Topological categories, Special morphisms, Limits and colimits, Factorizationsystems for morphisms, Reflective and coreflective subconstructs, Stability of constructions, Relation with completion and compactification theories.

(2) Exponential objects and Cartesian closedness, Existence of function spaces, relation to compatibility of constructions, Situation in TOP

(3) Paracompactness, Paracompactness for topological spaces, relation with star refinement axiom for open covers, Paracompactness in the setting of Nearness, Products of paracompact spaces.

See VUB: www.vub.ac.be > onderwijs > master > wiskunde > master wiskunde

Class contact teachingLectures

Personal workExercises

Personal work

ExaminationOral with written preparation

Course notes

evacoleb@vub.ac.be

bob.lowen@ua.ac.be