The objective of the course is to familiarize students with the basic techniques and notions of General Topology. First of all it is intended that the students know the different ways of defining a topological space using techniques such as bases and subbases of open sets, neighborhood and neighborhood base systems. In the first lesson particular attention is also paid to the study of metric spaces. Next, basic topics of General Topology are studied, such as continuity of functions, construction of derived topological spaces (products and quotients), compactness and connectedness.



The subject aims for students to start their knowledge in topology, studying the basic structures needed in many other subjects belonging to the area of Geometry and Topology and also Mathematical Annalysis.

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 Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • See GUIDELINES (%): 100

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%

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

Web addresses

Topology without tears
http://www.topologywithouttears.net/

Topology Atlas
http://at.yorku.ca/topology/