Here we give details to what material is covered in each week, and provide references to self-study material and links to videos.

Week 0 (recommended to self-study before the course)

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

The lecture note Section 0 covers most of this topic, but you can also refer to your favorite textbook or youtube-video if you need to brush up on your knowledge about countability.

Week 1

Session 1 (Tue 25.2) NOTE: Material presented here is recommended to read or watch before the session, and this is the same for all future sessions. Feel free to combine both reading the lecture notes and watching videos if you like.

Lecture notes: Sections 1.1-1.3.2. Quick link to lecture notes: PDF and to video playlist: Youtube

Definition of a topological space: Youtube

Relative/subspace topology: Youtube

Closed sets. Interior, exterior, boundary: Youtube

Session 2 (Wed 26.2)

Lecture notes: Sections 1.3 and 1.4

Closure and accumulation points: Youtube

Basis and subbasis: Youtube

Week 2

Session 3 (Tue 4.3)

Lecture notes: Section 2.1

Sequential convergence and Hausdorff spaces: Youtube

Cluster points, First-countability: Youtube

Session 4 (Wed 5.3)

Lecture notes: Sections 2.2-2.4

Continuous functions: Youtube

Sequential continuity, homeomorphisms etc: Youtube

Spaces of functions: Youtube (This topic is optional)

Week 3

Session 5 (Tue 11.3)

Lecture notes: Sections 2.5-2.6

Inducing and coinducing a topology: Youtube

Product topology: Youtube

Session 6 (Wed 12.3)

Lecture notes: Sections 3.1-3.1.2

Compactness: Youtube

Some results & Local compactness: Youtube

Week 4

Session 7 (Tue 18.3)

Lecture notes: Section 3.2

Connectedness: Youtube

Connected components etc: Youtube

Path-connectedness: Youtube

Session 8 (Wed 19.3)

Compactification: Youtube and lecture notes Section 3.1.3

Separation axioms: Youtube and lecture notes Section 4.1

Week 5

Session 9 (Tue 25.3)

Lecture notes: Section 4.2

Countability axioms: Youtube

Lindelöf spaces, density, and separability: Youtube

Session 10 (Wed 26.3)

Lecture notes: Section 4.3-4.5

Urysohn's lemma: Youtube

Urysohn metrization theorem: Youtube

Week 6

Session 11 (Tue 1.4)

Lecture notes: Section 5.2-5.3

Topological vector spaces: Youtube

Weak topology: Youtube

Session 12 (Wed 2.4)

Lecture notes: Section 5.4

Tychonoff's theorem: Youtube

Banach-Alaoglu theorem: Youtube