# General Topology

## Objectives

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

##### Code

11585

##### Credits

6.0

##### Responsible teacher

João Pedro Bizarro Cabral

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

Differential calculus in R^{n}. Basic notions of linear algebra.

Basic notions of metric spaces.

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

### Bibliography

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.

Teresa Monteiro Fernandes. Topologia algébrica e Teoria Elementar dos Feixes. Textos em Matemática, Faculdade de Ciências da Universidade de Lisboa

### 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

Knowledge assessment is carried out through Continuous Evaluation or Exam Evaluation Examination, presential. The Continuous Assessment consists of two tests and a grade in class.

**Grade In Class (AEA)**

For each week, a list of proposed exercises for students to work on before practical classes will be made available in advance. In the practical component of the classes, students make presentations based on this work, and at the end of the semester a grade of 0, 1 or 2 is attributed to each student.

**Continuous evaluation**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

During the semester two tests will be carried out with a duration of 1 hour 30 minutes. Each test is rated up to a maximum of 20 values.

Let *CT* be the simple arithmetic mean of the two tests rounded up to the units and *NF*the minimum between 20 and CT + AEA. The student obtains approval in the course if NF≥10. If NF≤16, the student is approved with the final classification NF. If NF≥17, the student can choose to stay with the final classification of 16 values or take a complementary test to defend his grade.

**Exam**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

All students enrolled in the discipline who have obtained Frequency or are exempt from it, and who have not obtained approval in the Continuous Evaluation, can take the appeal exam. Students can choose to repeat one of the 1 hour and 30 minute tests. If the student chooses to repeat one of the tests, the classification is calculated as in the case of Continuous Evaluation. If the student takes the appeal exam, let ER be the exam classification, rounded to the nearest integer, and NR the minimum between 20 and ER+ AEA. The student obtains approval in the course if NR≥10. If NR≤16, the student is approved with the final classification NR. If NR≥17, the student can choose to stay with the final classification of 16 values or take a complementary test to defend his grade.

**Grade improvement**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

Students have the right to improve their grades, upon enrollment within the deadlines, at the time of exam. In that case, they can take the 3 hour Exam or repeat one of the 1 hour and 30 minute tests as described in the previous paragraph. In the event that a student makes a grade improvement having obtained approval in a previous semester, he can only take the 3 hour exam.

**Logistics**

In order to rationalize the resources of FCT (facilities, teaching staff and non-teaching staff), only students who register for the purpose through CLIP, during the period stipulated therein, may take any of the tests.

If, at the time of the exam, the student chooses to repeat one of the tests, he / she must register for this test, otherwise he / she will take an appeal exam.

Only students who, at the time of the exam, carry an official identification document, containing a photograph (for example, Citizen Card, Identity Card, Passport, some versions of Student Card) and blank exam notebook.

**Final considerations**

In any omitted situation, the Knowledge Assessment Regulation of the FCT-UNL applies.

## 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.