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Het doel van het vak Inleiding Topologie is dat de student vertrouwd raakt met de fundamentele begrippen van de topologie, met name open en gesloten verzamelingen, compactheid, samenhangendheid, compactificatie, metriseerbaarheid. Na afloop van de cursus moet de student met deze begrippen kunnen werken en ze kunnen toepassen in wiskundige situaties.
Het vak Inleiding Topologie is een gebonden keuzevak voor wiskundestudenten. Het vak is essentieel voor veel richtingen in de zuivere wiskunde, maar is voor een brede groep studenten aan te raden, ook in de richtingen in maat- en integratietheorie, differentiaalvergelijkingen en dynamische systemen. Zie voor meer informatie over de studiepaden de studentenwebsite.
Leerdoelen:
The following topics will be discussed in the course.
- 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.
After attending 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 student is able to:
- manipulate with the basic concepts of topology; show the axioms for a topology; prove that a given function is continuous, or that a sequence is convergent; to compute in examples interiors, closures and boundaries; to write proper proofs using these concepts;
- manipulate with explicit examples, perform gluings or collapsing a subspaces (as an example of quotients);
- use the various topological properties in order to distinguish 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;
- manipulate with quotients 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;
- work with the one point compactifion;
- use paracompactness and partition of unity;
- use normality and Urysohns lemma on the existence of separating functions;
- understand Urysohn and Smirnov metrizability theorems.
Onderwijsvormen:
Each week there are two lecture and two exercise classes, each of two hours. There will be numer of exercises for instruction in class, and a number of homework exercises (mandatory).
Toetsing:
There will be a total of 6 mandatory homework exercises (one per week). 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 retake exam will be a 3 hour written exam, leading to grade E with 1 decimal of accuracy.
The final grade F will be obtained from E by rounding off to an integral number up to 6, and to a half integer above 6.
Herkansing en inspanningsverplichting:
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.
Taal van het vak:
The language of instruction is English or Dutch, depending on the lecturer in a specific year.
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