MS-E1542: General Topology

Description

TLDR

Introduction to Topology, could be taken with no previous knowledge.

General

General Topology is a rigorous introduction to the theory of general topological spaces. The course develops the fundamental notions of topology, continuity, convergence, compactness, connectedness, and separation, and proceeds to more advanced topics such as metrization, quotient spaces, homotopies, topological vector spaces, and weak topologies.

Topology is a fundamental branch of mathematics with applications throughout mathematics and other disciplines. The course begins with the basic building blocks of topological spaces and develops the theory through proofs and exercises.

Learning outcomes (official)

• developing the logical reasoning skills to write rigorous proofs in mathematics
• ability to describe fundamental topological concepts such as continuity, convergence, and compactness
• applying basic definitions and results in topology to evaluate the properties of various topological spaces
• building a solid foundation in set theory and its operations

Content (official)

• Topological spaces
• Continuity and convergence
• Product topology
• Compactness and Connectedness
• Separation and countability axioms
• Urysohn’s lemma and metrization theorem
• Topological vector spaces

Course material

Official material

The main course material consists of lecture notes:

Lecture notes, version 2.4.2024

There is also a video lecture series:

Course video playlist on YouTube

The course Timeline indicates which sections of the notes and which videos are covered in each session.

Extra material

The Finnish book Topologia II written by Väisälä is great.

Contents and workload

The course has two exercise sessions per week with obligatory attendance. The exam was relatively easy compared to other master’s level courses.

Overall workload

Weekly contents

Detailed weekly topics

Week 0

Recommended self-study before the course:

• Countable and uncountable sets
• Proof that the rational numbers are countable
• Proof that the irrational numbers are uncountable

Week 1

• Definition of a topological space
• Relative/subspace topology
• Closed sets
• Interior, exterior, and boundary
• Closure and accumulation points
• Basis and subbasis

Week 2

• Sequential convergence
• Hausdorff spaces
• Cluster points
• First-countability
• Continuous functions
• Sequential continuity
• Homeomorphisms
• Spaces of functions

Week 3

• Inducing and coinducing a topology
• Product topology
• Compactness
• Local compactness

Week 4

• Connectedness
• Connected components
• Path-connectedness
• Compactification
• Separation axioms

Week 5

• Countability axioms
• Lindelöf spaces
• Density
• Separability
• Urysohn’s lemma
• Urysohn metrization theorem
• Tietze extension theorem

Week 6

• Topological vector spaces
• Weak topology
• Tychonoff’s theorem
• Banach–Alaoglu theorem

Practicalities

The course has two main weekly sessions.

The sessions primarily consist of solving exercises and proving results related to the week’s topics. Students can receive assistance from the teacher and assistant during the sessions.

Students are expected to study the assigned reading and/or video material before each session.

Exercise credits form the majority of the course grade. Students who cannot attend the weekly sessions can alternatively obtain the corresponding credits by submitting written solutions.

There are also 2–3 homework exercises per week.

Assessment

The course grade consists of:

The weekly exercises consist of:

• participation in the exercise sessions;
• alternatively, submitting written solutions to the session exercises;
• weekly homework assignments.

The final exam is a smaller component than the exercises, but a minimum exam score is required to pass the course. The exam was no more difficult than the course exercises.

Grading

The minimum requirements were:

To pass, students needed at least 50% of the total score and 50% of the exam score.

The exact grade boundaries could be adjusted slightly after the exam if the exam proved unexpectedly difficult.

Final exam

The following concepts were explicitly listed as material whose definitions students were expected to remember:

• Finite, countable, and uncountable sets
• Topology / topological space
• Open set
• Neighbourhood
• Subspace / relative topology
• Interior
• Exterior
• Boundary
• Closed set
• Closure
• Basis
• Subbasis
• Converging sequence / limit of a sequence
• Hausdorff space
• First-countable space
• Continuous function
• Open and closed map
• Homeomorphism
• Induced topology
• Product topology
• Projection map
• Open cover
• Compact space
• Locally compact space
• Connected space
• Regular space
• Normal space
• Second-countable space
• Dense set
• Separable space
• Topological vector space

Related courses

Official prerequisites

MS-C1541: Metric Spaces

Additional prerequisites

No additional prerequisites. Unlike many other E-courses, this can be taken without major pain at bachelor’s level.

More like this

• Topologia II from University of Helsinki

Trivia

• The course places unusually strong emphasis on proving results and solving exercises during contact sessions rather than traditional lecture-style teaching.
• The course’s final topics go beyond the usual introductory topology curriculum, reaching topological vector spaces, weak topologies, Tychonoff’s theorem, and the Banach–Alaoglu theorem.
• The lecture notes and videos were produced specifically for this course.
• First lectured in 2024. The original course code was MS-EV0021.

Timestamp

Based on the 2024 version of the course, taught in period IV.