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# Metric and Topological Spaces

 Code 14781 Year 2 Semester S2 ECTS Credits 6 Workload TP(60H) Scientific area Mathematics Entry requirements Does not have. 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. Syllabus 1. METRIC SPACES1.1 Definition and first examples 1.2 Lebesgue sequence spaces1.3 Open balls and closed balls1.4 Open sets and closed sets1.5 Interior, exterior, boundary, closure and derived1.6 Continuous functions1.7 Sequences1.8 Complete metric spaces1.9 Completion of a metric space1.10 Banach’s fixed point theorem1.11 Baire’s theorem1.12 Compactness in metric spaces1.13 Connectedness in metric spaces2. TOPOLOGICAL SPACES2.1 Definition and examples2.2 Closed sets2.3 Bases and sub-bases2.4 Interior, exterior, boundary, closure and derived2.5 Continuous functions2.6 Subspaces2.7 Product spaces2.8 Separation axioms2.9 Compactness2.10 Connectedness 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 ediçãoLima, E. L. (2014). Elementos de topologia geral. Textos Universitários. Sociedade Brasileira de MatemáticaMunkres, J. R. (2000). Topology. Prentice Hall.Willard, S. (2013). General Topology. Dover Publications. Language Portuguese. Tutorial support is available in English.

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Mathematics and Applications
Last updated on: 2021-06-19

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