Topology

Kurssin kuvaus suomeksi täältä.

Objectives and Course Description

Topology is the study of forms, shapes and spaces using notions that do not depend on the notion of a distance, and which generalise the notion of space beyond that of a metric space. The course is a natural continuation of the Metric Spaces course.

The basic notions of topology are open sets, closed sets, continuous maps and convergence. You have already seen these in the Metric Spaces course, but here, we will define them for general spaces. We will see how the newly developed language can be very flexible; e.g. we define continuous maps in a very short and convenient fashion. We will also see examples of topologies, where it is necessary to use this more general language.

We will study the basic properties and axioms of topologies, and see some of the common ways of constructing topologies from others (relative, product, induced topologies).

The main learning objectives for the course are:

Understand the core concepts/definitions: topology, Hausdorff topology, relative topology, product topology and compact space
Be familiar with the Baire category theorem and some of its applications.
Have the ability to apply the concepts from the class to proofs in different contexts.
Has improved the ability to understand concepts in different areas of mathematics by employing concepts from this course.

We will see applications of the course topics to various other concepts, that you may encounter during your math studies, such as:

Convergence of functions (analysis, functional analysis).
The topology on the Riemann sphere (complex variables)
Topologies on simple manifolds/orbifolds, that arise as group quotients (differential geometry, algebraic topology).
Zariski topology in algebraic geometry (algebra, algebraic geometry).
Weak vs. strong topologies, product topologies (functional analysis, probability).
Word ultrametric for sequences. (number theory, fractal spaces, dynamics)

HW is available on moodle.

LEcture times

Lectures: Sylvester Eriksson-Bique

Wednesday+Thursday 25.10-15.12.2023

12:15-14:00 MaD 380

Office hours: Sylvester Eriksson-Bique

Tuesdays 16:00-17:00, I will be available in ratkomo.

Available upon request at other times.

Demo sessions:

Wednesday 14:15-16:00

Exam:

13.12.2024, 8-12

Location TBD

Syllabus

This is the first time I give this course and the first time it is given with the proposed topics. Thus the following schedule is only a rough estimate, and we may move a bit slower or faster. Contact me if the pace is not ok for you.

Week 1: Definition of topology, Relative topology, Continuity, Convergence, Open, Closed sets, boundary, exterior, interior, separation axioms.

Objective: Learn to define a topology, start to use the definitions to prove basic properties. Give examples of topologies on finite sets.

Application: Zariski topology

Week 2: Closure, Density, Separability, finer and coarser topologies, more on Hausdorff axioms.

Objective: Understand that the same space can be equipped with multiple topologies. Understand the role of countability and assuming separability.

Week 3: Compact Spaces, Complete metric spaces, G-delta sets , F-sigma sets, meager and comeager sets, and Baire Category

Objective: Apply the open cover definition of compact space. Recognize that completeness is not invariant under homeomorphisms. Use Baire category to conclude that rational numbers can not be made into a complete metric space.

Application: definition of topology on Riemann Sphere

Week 4: More applications of Baire Category, Concept of a basis for a topology, Neighborhood basis,

Objective: Understand the definition of relative topology, and distinguish between the relative and original topologies. Define a topology using a neighborhood basis.

Week 5: Product topology, Infinite product topologies, Induced topology, categorical thinking

Objective: Define topologies using a neighborhood basis, study the product topology. Define topologies as the coarsest/finest topology with the property that given maps are continuous. Understand continuity via diagrams. Define and distinguish topologies of uniform and pointwise convergence. Understand that the latter is not metrizable.

Week 6: Quotient topologies and glued spaces

Objective: Define topologies for torus, Klein bottle, Möbious strip and lens space. Define topologies for glued spaces. Begin to understand totally discontinuous group actions.

Literature

J. Munkres: Topology (2nd ed.)
S. Willard: General topology

Lecture notes in Finnish from Jouni Parkkonen in Moodle.

There are many sources for the material for this course. One is this one.

The chapters 5,6,7,9,10,11,12,13 together with parts of 3 and 14,15 cover nearly all of the contents. The order is not precisely the same though. We will end by discussion of some quotient spaces and topologies. These we will give notes on closer to that week.

Additional references:

I will write weekly summary notes, which include references and the plan for the week.

Grading

The course can be completed in one of two ways, and you will receive the better of :

HW+Portfolio (40 %) and a final exam (60%)
Final exam (100%)

Out of the above two options, the first is highly recommended. In any case, you will receive the score which is better of the two.

You must complete the final exam to get credit. On the final exam, you must complete one problem nearly completely correctly to pass the exam.

If for a justified reason you are unable to complete some of the requirements, please contact me, and we may find a compensatory way.

The course is 5 cr, and graded 0-5. As a base level, I will use the following rubric, which however can be adjusted in a way that increases grades.

90 %, 5
80 %, 4
70 %, 3
60 %, 2
50 %, 1
<50 %, 0/Fail

HW+Portfolio

The course has 6 HWs and 1 Portfolio, each of which is worth 20 points, and consist of a written portion (10pt), a group portion (5pt) and portfolio (5pt). The group portion can be done alone. From the portfolio, you can get an additional 0-5 bonus points for exceptional performance.

Each week, there are written problems (submitted as part of the HW), group problems (done together in DEMOs or submitted in writing) and a portfolio problem, which you complete either in DEMO’s or partially on your own.

Each student should complete the written portion on their own, and think ahead of time about the group portion. You should attempt to solve the group portion, and write at least a sketch of your ideas – but you can complete the solution in the sessions. In the demo sessions, the group problems are discussed. You may work on the problems alone, however, if you wish.

HW should be returned in the demo sessions and to get the credit for the group portion, you should participate in the discussion section. But, no-one can force you to talk in the session. If you are unable to participate, you may submit written solutions to all problems which are graded on completeness and correctness.

Portfolio containing all work for each week is submitted by the end of the course, by December 14th the latest. You can also bring your portfolio as printed with you to the final exam.

Generally, no late assignments will be accepted. If you are sick, or otherwise unable to complete an assignment, please contact the instructor in advance, and discuss alternate options.

The lowest HW score is dropped and ignored in the grading.

Inclusivity

The course should be an encouraging and open space for discussion, learning and the exchange of ideas. Every participant should be able to join fully and comfortably. If you find challenges with this, or if any course policy is difficult for you, please discuss these with the instructor. Also bear in mind, that the university has resources and support in the event of harassment or of personal difficulties. I can assist you in identifying such resources.

If you need a religious exemption, or other compelling exemption, please contact me at least 1 week prior to the requests effect. I may not be able to accommodate requests that arrive late.