| Code |
15374
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| Year |
3
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| Semester |
S2
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Does not have.
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Learning outcomes |
In this curricular unit the students are expected to be familiarized with the concepts, principles and methods of metric spaces and topological spaces and be able to apply them to other areas of mathematics.
At the end of the curricular unit the students should be able to:
- apply the basic concepts of metric spaces and topological spaces;
- recognize the importance of complete metric spaces;
- understand the proof and apply Banach's fixed-point theorem;
- understand the proof and apply Baire's theorem;
- illustrate the definitions with examples;
- understand the proofs of basic results on separation, compactness and connectedness of topological spaces.
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Syllabus |
0. GENERALITIES
0.1 Image and preimage of a set by a function
0.2 Families of sets
1. METRIC SPACES
1.1 Definition and first exemples
1.2 Normed spaces
1.3 Open balls and closed balls
1.4 Open sets and closed sets
1.5 Interior, exterior, boundary, closure and derived
1.6 Continuous functions
1.7 Sequences
1.8 Cauchy sequences and complete metric spaces
1.9 Completion of a metric space
1.10 Banach’s fixed point theorem
1.11 Baire’s theorem
1.12 Compactness in metric spaces
1.13 Connectedness in metric spaces
2. TOPOLOGICAL SPACES
2.1 Definition and examples
2.2 Closed sets
2.3 Bases and sub-bases
2.4 Interior, exterior, boundary, closure and derived
2.5 Continuous functions
2.6 Subspaces
2.7 Product spaces
2.8 Separation axioms
2.9 Compactness
2.10 Connectedness
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Main Bibliography |
– Buskes, G., & van Rooij, A. (2012). Topological Spaces: From Distance to Neighborhood. Springer.
– Conway, J. B. (2014). A Course in Point Set Topology. Springer.
– Croom, F. H. (2016). Principles of Topology. Dover Publications.
– Engelking, R. (1989). General Topology. Heldermann Verlag.
– Giles, J. R. (1987). Introduction to the Analysis of Metric Spaces. Cambridge University Press.
– Lima, E. L. (2017). Espaços Métricos. Projecto Euclides. Instituto de Matemática Pura e Aplicada. 5.ª edição
– Lima, E. L. (2014). Elementos de Topologia Geral. Textos Universitários. Sociedade Brasileira de Matemática
– Munkres, J. R. (2000). Topology. Prentice Hall.
– Willard, S. (2013). General Topology. Dover Publications.
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Teaching Methodologies and Assessment Criteria |
The curricular unit is organized in theoretical-practical lessons. At the beginning of each lesson the teacher makes a theoretical presentation of the subject and then the students solve exercises from worksheets provided by the teacher.
The continuous evaluation consists of two written tests of 10 points each. If T1 and T2 are the grades of the first and second tests, respectively, the final grade is calculated as follows:
a) if T1+T2 is less than 17.5, the final grade will be the rounding of T1+T2;
b) if T1+T2 is greater than or equal to 17.5, the student must do an oral exam; to the oral exam is given a grade PO between 0 and 20 points; the final grade will be the rounding of max{17,(T1+T2+PO)/2}.
The students with a final grade greater than or equal to 10 points will pass the course.
The evaluation by final exam consists of an exam with the maximum value of 20 points. The students with a classification greater than or equal to 10 points will pass the course.
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Language |
Portuguese. Tutorial support is available in English.
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