SPECIFIC COMPETENCIES



M02CM11 - Understand the basic concepts, methods, results and proofs related to Topological spaces and Metric spaces.

M02CM12 - Assimilate the concepts of Continuity, Compacness and Connectedness.

M02CM13 - Recognize topological structures in concrete examples.

M02CM14 - Construct examples of topological spaces using the notions of subspace, product space and quotient space

M02CM15 - Use convergence of sequences to study continuity and compacness.





LEARNING OUTCOMES



- Recognize topological structures in concrete examples.

- Construct examples of topological spaces using the notions of subspace, product space and quotient space.

- Use convergence of sequences to study continuity and compacness.

1. TOPOLOGICAL SPACES: Topology. Open and closed sets. Base and subbase of a topology. Neighbourhoods. Neighbourhood bases. Distance. Metric spaces. Open and closed balls.



2. SUBSETS IN TOPOLOGICAL SPACES: The interior of a set. The closure of a set. Accumulation points and isolated points. The derived set. The boundary of a set.



3. CONTINUITY: Continuous functions. Homeomorphisms. Topological properties. Sequences in metric spaces: convergence and sequential continuity.



4. CONSTRUCTION OF TOPOLOGICAL SPACES: Subspaces. Combined functions. Embeddings. Product topology. Projections. Quotient topology. Identifications.



5. COMPACTNESS: Compact spaces and compact subsets. Products of compact spaces. Sequential compactness. Compactness in Hausdorff spaces.



6. CONNECTEDNESS AND PATH CONNECTEDNESS: Connected spaces and connected subsets. Connected components. Paths in topological spaces. Path connectedness. Path-components.

The theoretical sessions will be presented in the lectures, following the basic references contained in the Bibliography and the mandatory material. These lectures will be complemented with problem-solving classes in the practical classroom work sessions, in which the knowledge acquired in the theoretical classes will be applied. Finally, in the seminar sessions, students will take a more active role and will develop and discuss representative examples/exercises of the contents of the subject.

CONTINUOUS EVALUATION



Written exam (weight: %70-%85)

Evaluation criteria:

- Accuracy on definitions and reasoning.

- Appropriate use of mathematical language.

- Correct methods of reasoning, with clear and well organized explanations of the arguments and the intermediate steps.



Seminars (weight: %5-%10)

Evaluation criteria:

- Correct answers and appropriate use of mathematical language.

- Clear reasoning.

- In oral presentations, accuracy and order.



Resolution of written exercises (weight: %10-%20)

Evaluation criteria:

- Correct answers and appropriate use of mathematical language.

- Clear reasoning.

- Accuracy and order in the exercises delivered.



FINAL EVALUATION (in case of renouncing the continuous evaluation)



Written exam: 100%

Classroom notes. Proposed exercise list.

Basic bibliography

Theory



R. AYALA, E. DOMINGUEZ y A. QUINTERO; Elementos de Topología General, Addison-Wesley Iberoamericana, 1997.

J. R. MUNKRES, Topología, Prentice Hall, 2002.

S. WILLARD, General Topology, Dover Publications Inc, 2004.



Problems and exercises



G. FLEITAS MORALES Y MARGALEF ROIG, Problemas de Topología General, Alhambra, 1980.

G. FLORY; Ejercicios de Topología y Análisis, Reverté, 1978.

E.G. MILEWSKI, Problem solvers. Topology, Research & Education Association, 1994.

In-depth bibliography

I. ADAMSON; A General Topology Workbook, Birkhäuser, 1995.
E. BURRONI, J. PENON, La géometrie du caoutchouc. Topologie, Ellipses, 2000.
L. A. STEEN y J. A. SEEBACH, Counterexamples in Topology, Dover, 1995.
O. YA. VIRO, O. A. IVANOV, N. YU. NETSVETAEV y V. M. KHARLAMOV, Elementary Topology. Problem Textbook, AMS, 2008.

Journals

Americal Mathematical Monthly