- The intuitive notion of "space" (+ definition of metric spaces) and standard examples (spheres, Moebius band, torus, Klein bottle, projective space etc).

- The abstract definition of topological space; first examples; metric topology; metrizability; Hausdorffness, separation axioms and normal spaces; subspace topology.

- Neighborhoods; continuity; homeomorphisms; embeddings; converegence and sequential continuity; basis of neighborhoods and 1st countability.

- Inside a topological space: interior, closure, boundary.

- Quotient topology; special quotients (e.g. quotients modulo group actions; collapsing a subspace to a point; cylinders, cones, suspensions).

- Product topology, bases for topologies, generated topologies.

- Spaces of functions; pointwise, uniform, uniform on compacts convergence; completeness with respect to the sup metric.

- Connectedness, path connectedness, connected components.

- Compactness, basic properties, compactness in metric spaces (characterizations in terms of completeness and total boundedness), finite partitions of unity; sequential compactness.

- Local compactness; the one-point compactification.

- Paracompactness and arbitrary partitions of unity. Criteria for paracompactness.

- Urysohn's lemma, the Urysohn metrizability theorem, the Smirnov metrizability theorem.

- The Stone-Weierstrass theorem.

- The algebra C(X) of continuous functions on a compact space and its C^*-algebraic structure. The Gelfand-Naimark theorem.

Knowledge and insight information
Affter following the course, the student knows/understands:

     - the standard examples (spheres, tori, Moebius bands, projective spaces) and manipulations with them (gluing, etc).

     - the basic notions of topology: the abstract notion of topological space, convergence, continuity, homeomorphisms, interior, closure,.

     - the standard constructions of topological spaces: metric topologies, induced topologies, quotient topologies, product topologies, generated topologies.

     - the most important topological properties: Hausdorffness, connectedness, compactness, local compactness.

     - the usefulness of compactness for proving embedding results; characterizations of compactness in metric spaces.

     - several metrizability results.

     - the statements of the Stone-Weierstrass theorem and of the Gelfand-Naimark theorem.

Skills
The student is able to:

     - manipulate with the basic concepts of topology; be able to show the axioms for a topology; be able to prove that a given function is continuous, or that a sequence is convergent, be able to compute in examples interiors, closures and boundaries. Be able to write proper proofs using these concepts.

     - be able to manipulate with explicit examples, perform gluings or collapsing a subspaces (as an example of quotients).

     - be able to use the various topological properties in order to distinguish that certain topological spaces (proving that they are not homeomorphic). Example: a circle is not homeomorphic to a bouquet of two circles because, after removing any point from a circle the result is connected, while the corresponding property is not true for the bouquet.

     - be able to manipulate with quotient and to compute quotients. Be able to show that a given map is an embedding (e.g by using compactness).

     - use compactness and sequential compactness.

     - be able to prove density results, by applying the Stone-Weierstrass theorem.

     - in examples., be able to compute the characters and the spectrum of a given algebra.

Onderwijsvormen: Each week there is a lecture and an exercise class. There will be some homeworks (compulsory), as well as some more difficult "bonus exercises".

Exam+ Final mark:

There will be two written exams. For each one of them the mark should be at least 5. Their average makes the final exam mark, called E.

 There will also be take home exercises (one per week, or per two weeks, depending on the number of students). For each exercise, the deadline for solving it is two weeks, and a mark will be given. The average of the marks for the exercises will make the final exercises mark, called W. It should be at least 5.

 There will also be a few bonus exercises, which are more difficult, and which are not compulsory. For each such exercise one can earn some points (usually 0.25 or 0.50 per exercise). These points add together to the total bonus points, called B.

 The final mark will be min{10, B+ (3E+ 2W)/5}, and it should be at least 6.

 

There will also be a retake (hertentamen). For the retake, the homework and the bonus exercises do not count (hence you may expect that the retake will be more difficult than the other two exams).

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