Topology
| Keywords |
| Classification |
Keyword |
| OFICIAL |
Mathematics |
Instance: 2025/2026 - 1S 
Cycles of Study/Courses
Teaching Staff - Responsibilities
Teaching language
Portuguese
Objectives
Acquisition of fundamental knowledge of General Topology necessary for Analysis, Differential Geometry, Algebraic Geometry and Algebraic Topology. Topological group actions. Topics of Algebraic Topology: path homotopy, fundamental group, covering spaces.
Learning outcomes and competences
The student should become able to apply knowledge of Topology to different areas of Mathematics.Working method
Presencial
Program
Reminder of concepts and theorems of normed and metric spaces. Topological spaces, continuous maps, first countable spaces, convergente of nets, categorial construction of topological spaces: subspaces, products, coproducts, quotients of topological spaces. Initial and final topologies. Hausdorff spaces. Second-countable Spaces. Separable spaces. Connected and arcwise connected spaces. Compact spaces, sequentially compact spaces. Regular and Normal spaces. Urysohn and Tietze theorems. Urysohn metrizability theorem. Locally compac spaces, Alexandroff compactification. Paracompact spaces, partitions of unity subordinated to a covering. Proper maps. Topological groups and topological group actions. Path homotopy, Fundamental group, Covering spaces.
Mandatory literature
Jacques Dixmier; General Topology, Springer, 1984. ISBN: 978-1-4757-4032-5
Willard , Stephen;
General topology. ISBN: 0-201-08707-3
William Fulton;
Algebraic topology. ISBN: 0-387-94327-7
Complementary Bibliography
Glen E. Bredon;
Topology and geometry. ISBN: 0-387-97926-3
Teaching methods and learning activities
The contact hours consist of theoretical and practical lessons allowing the instructor torganize and manage the time avaible for presenting the subject matter, solving exercises and student presentations.
keywords
Physical sciences > Mathematics > Geometry
Evaluation Type
Evaluation with final exam
Assessment Components
| designation |
Weight (%) |
| Exame |
100,00 |
| Total: |
100,00 |
Amount of time allocated to each course unit
| designation |
Time (hours) |
| Estudo autónomo |
120,00 |
| Frequência das aulas |
42,00 |
| Total: |
162,00 |
Eligibility for exams
The attendence of classes is not mandatory.
Calculation formula of final grade
The final mark is the mark obtained in the final exam.
Special assessment (TE, DA, ...)
By one single oral or written exam.
Classification improvement
By exam in accordance with the regulations. Results from distributed assesment in previous academic years cannot be used.