Does not have.
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.
1. METRIC SPACES
1.1 Definition and first examples
1.2 Lebesgue sequence 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
1.8 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
2.7 Product spaces
2.8 Separation axioms
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.
Portuguese. Tutorial support is available in English.