Topology

Code: MAT14226L

6 ECTS

Duration: 15 weeks/156 hours

Scientific Area: Mathematics

Teaching languages: Portuguese

Languages of tutoring support: Portuguese

Presentation

In this discipline, taught in the 2nd year, it is intended that the student develops the topological abstract thinking necessary to be able to work with modern mathematical structures.

Sustainable Development Goals

Educação de qualidade - Assegurar a educação inclusiva, e equitativa e de qualidade, e promover oportunidades de aprendizagem ao longo da vida para todos.

Learning Goals

To develop the abstract topologic mode of thinking necessary to be able to work with the structures of the modern mathematics. Since the topological methods are very important for many disciplines, the students should acquire the basic knowledge of this material and capacities to apply it to various problems that can occur.

Topological spaces. Open and closed sets. Cluster points and closure. Density. Hausdorff axiom. Sequences and convergence. Continuous functions. Compact spaces. Comparison of topologies. Subspaces and product of spaces. Tikhonov theorem. Complete metric spaces. Completion. Uniform continuity. Connected spaces. Banach and Hilbert spaces.

Teaching Methods

Students will have at their disposal in Moodle all the material used in the classes, which consists of a presentation and initial discussion of the topics to be studied, followed by illustrative practical applications. The evaluation can be done by two processes (by frequency and / or by exam), each one carried out with the possibility of consulting the material produced by himself:
Assessment by Exam: The student will pass if, in one of the exams, he obtains a classification equal to or higher than 10 values.
Continuous Assessment: Two tests will be carried out, focusing on a part of the subject taught.
The classification of this component will be the average of the classifications obtained. The student will choose Continuous Assessment if he presents himself for the evaluation in the two frequencies and has, in each of them, a classification equal to or higher than eight values.