Introduction to topology

• 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.- one point compactifion
• paracompactness and partition of unity
• normality and Urysohns lemma on the existence of separating functions
• partitions on one
• Urysohn and Smirnov metrizability theorems
  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 mandatory weekly homework exercises,
  6 in total. These lead to a grade H with 1 decimal of accuracy.
  At the end of the course there will be a 3 hour written exam, leading to a grade E with 1 decimal of accuracy. The final grade F is be determined by F = max {(7E + 3H)/10, (17E + 3 H)/20}, rounded off to an integral number up to 6, and to a half integer above 6.
  The requirement for admission to the retake exam is: either a grade of 4 or 5 for the written exam, or a score lower for the written exam supplemented by at least 4 of the home work exercises, with average grade 6.

Become familiar with fundamental notion is topology, notably open and closed sets, compactness, connectedness, compactification, metrisability. After completing the course, the student is able to work with these concepts and apply them in a mathematical setting.

WISB102 Proofs in mathematics, WISB114 Analysis and WISB124 Introduction to groups and rings (preferably, but the necessary theory can be learnt during the course). See the courseplanner (cursusplanner.uu.nl) for the contents of those courses: select Faculty of Science and then the programme of the bachelor Mathematics of the most recent year.

https://osiris-student.uu.nl/onderwijscatalogus/extern/cursus?cursusid=236016&taal=en