Inleiding Topologie

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 verplicht vak voor wiskundestudenten.

Zie ook de website van het vak: https://webspace.science.uu.nl/~crain101/topology-2024/.

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 a number of exercises for instruction in class, and a number of homework exercises (mandatory).

Toetsing:
There will be mandatory weekly homework exercises, whose solution is to be written down in the class (not at home). If n is the total number of homeworks, the average of your best n-1 marks will be denoted H and called the overall homework mark, with two decimals of accuracy.

At the end of the course there will also be a 3 hour written exam, leading to a grade E with two decimals of accuracy.

There may also be occasional bonus exercises (hard ones), that add up to a certain number of points B.

The final grade F is F= min{10, (6 E+ 4 HW)/10 + B} rounded off to the closest integer if F< 6, and to a half integer otherwise. The requirements for passing the exam are: H and E have to be at least 5 (before rounding) and F has to be at least 6.

The grade rules can still change, due to AI.

Herkansing en inspanningsverplichting:
The same rules apply for the retake.

Taal van het vak:
The language of instruction is English.

Zie onder vakinhoud.

WISB102 Bewijzen in de Wiskunde, WISB114 Analyse en WISB124 Inleiding groepen en ringen (bij voorkeur, maar de nodige kennis kan tijdens de cursus geleerd worden). Zie de cursusplanner (cursusplanner.uu.nl) voor de inhoud van deze vakken: selecteer Faculteit Betawetenschappen en vervolgens het programma van de bachelor Wiskunde van het meest recente jaar.

• Dictation There are revised lecture notes (dictaat), divided into two parts the theory, and exercises. They are available via A-Eskwadraat and can also be found on the website of the course: https://webspace.science.uu.nl/~crain101/topology-2024/