General Topology


The aim of this course is to provide students with general knowledge of topology, illustrated with numerous examples, also acting as an introduction to Algebraic Topology and Homotopy Theory. Students should understand and know how to demonstrate the fundamental results on topological spaces, continuous functions, compactness, connectedness and separation/countability axioms. In Algebraic Topology, students will learn the concept of the fundamental group of a topological space and how to compute it in simple cases, using the Van Kampen Theorem or covering spaces.

General characterization





Responsible teacher

Ana Cristina Malheiro Casimiro


Weekly - 4

Total - Available soon

Teaching language



Differential calculus in Rn. Basic notions of linear algebra.

Basic notions of metric spaces.

Basic notions of equivalence relations, groups, rings, fields, morphisms and isomorphsm.


John M. Lee: Introduction to Topological Manifolds, Second Edition, Graduate Texts in Mathematics 202, Springer, 2011 (principal book)

James R. Munkres: Topology. Prentice Hall (2000)

Armstrong, Mark Anthony: Basic topology. Corrected reprint of the 1979 original. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1983.

Gamelin, Theodore W.; Greene, Robert Everist: Introduction to topology. Second edition. Dover Publications, Inc., Mineola, NY, 1999.

Hatcher, Allen: Algebraic topology.  Cambridge University Press, Cambridge,2002.

Teaching method

The course will have 3 hours classes dedicated to theory and practice and also 1 hour of student support in class. Students must, by themselves, solve advance exercizes, some of them being the proof of some essentials resuts.

Evaluation method

The course will have 4 hours classes dedicated to theory and practice. Students must, by themselves, solve advance exercises, some of them being the proof of some essentials resuts.

Students must solve a designated list of exercises regularly and, give lectures about the theoretical part and the exercises.

The final grade will be the arithmetic average of the grades obtained in the lists of exercises which counts 40% and lectures that will count others 40% of the course grade. There will be a written test, of 1h30m length, in person, with consultation, whose grade will count 20% for the course grade.

Subject matter

1)Topological spaces. Base and sub-base. Continuous functions. Homeomorphism. Properties of topological spaces.

2)Connectness. Path connectness. Connected components.

3)Compact spaces. Sub-base lemma. Cartesian product of topological spaces.Tychonoff theorem. Compact metric spaces. Lebesgue number. Local compact spaces.

4)Countability axioms. Separation Axioms. Urysohn lemma. Tietze extension theorem.

5) Quotient topology.

6) Path homotopy. Fundamental group. Base point independence. Fundamental group of the circle. Categories and functors. Homotopy equivalence. Covering spaces. Lifting paths and homotopies.

Additional topics: Manifolds. Spaces of Functions. Spaces of Baire. Classification of surfaces. Classification of covers.


Programs where the course is taught: