Topology I - point set topology

Welcome to Math 220  (Topology I - point set topology)

All the relevant course information and materials can be found here (on Moodle).

• Lectures take place every Tuesday from 10h15-12:00 in ELA 1
• Exercise classes will be Tuesday 13h15-15:00 in MA A1 10 and MA A3 31

Below, you will find the "ED discussion" for this class.

The first lecture is on Tuesday, 10/09, at 10h15 in ELA 1. In terms of content, the first lecture will be an overview of the course, and we will also discuss formalities.

The first exercise class is on Tuesday, 10/09, at 13h15. The classrooms are MA A3 31 and MA A1 10. Below is the first worksheet, which is more of a review/warm-up.

I am looking forward to teaching this class and meeting all of you. I hope we will all have some fun this semester!

This week we will learn about the concept of a topology, and recall that of a metric. In particular, we will discuss how any metric induces a topology.
Moreover, we will further look at some intrinsic notions: interior, closure, boundary of a set.

The content corresponds to Section 1.1 in the lecture notes.

This week we start our discussion on continuity. We will cover Sections 1.2 and 1.3 in the notes.

The topics for this week correspond to Section 1.4 in the notes.

Here are two suggestions for some extra reading:

• Sections 6 to 10 in Chapter 3 of these notes by Pete L. Clark (freely available on his webpage).
• Paragraph 22 in Chapter 2 of Munkres' book has a good (meaning one I like) treatment of quotient topology.

This week, we start Chapter 2. The content corresponds to Section 2.0 and the beginning of 2.1 in the notes.

Here is a suggestion for extra reading.

• These notes by Keith Conrad (freely available on his webpage) contain a detailed exposition of two examples of topological spaces that are connected but not path-connected. 

This week, we will finish discussing  Section 2.1 in the notes and start with 2.2.

This week, we will finish Section 2.2 in the notes.

I will be travelling, and Dr. Nikolaos Tisakanikas will give the lecture in my place.

This week we will cover Sections 3.0, 3.1 and 3.2 in the notes.

This week we will discuss Sections 3.3, 3.4 and 3.5 in the notes.

In the notes, the material for this week corresponds to Sections 4.1 and 4.2.

This week, we will cover sections 4.3 and 4.4 in the notes.

We will cover sections 4.5 and 4.6 in the notes.

Below you find a list of exercises to concentrate on for the exam, and a way to conceptualize the material. 
Additionally, I suppose all the questions we have discussed in class could be a pretty useful way to check whether you have understood the course.

For the first session on Tuesday 15/09 at 10h15 AM we meet on ZOOM to discuss the contents and the format.
The link is here and pass is METTOP20

The first exercise class also takes place Wednesday 10h15. Same ZOOM link, but also PIAZZA live Q & A, for the 1/3 on campus room CE1 105.
The sheet will be up here by Tuesday evening.

This week we will look at natural ways to construct topologies from old ones via subsets, products and disjoint union.
As every week, there will be a Q&A session in ZOOM at 11h40.

This week we start a new chapter: we will see how the notion of connectedness can be formalized in the realm of topological spaces.
As every week, there will be a Q&A session in ZOOM at 11h40.

We continue with connectedness - with some of its main properties and show that connectedness is indeed a weaker notion than that of path-connectedness.
We also start discussing simply-connectedness.

As always, I meet you on ZOOM at 11h40 for a Q & A.

This week we discuss the fundamental group - a really beautiful object.
There is a ZOOM Q & A as always, but let's start this time at 11h20 to also go through the QUIZ.

We continue with beauty and start with the notion of compactness.
I meet you at 11h40 on ZOOM on Tuesday to take your questions

This week, we will start a new chapter on metric spaces.
I will also prepare a quiz on chapters 2 and 3.

PS! I believe you can again give feedback through the IS-academia, please do so independently whether you like or don't like the course - it is helpful to have your feedback in any case.

We are continuing our study of metric spaces and see how the notion of compactness behaves in this setting - they behave very nicely!
We will also introduce the notion of completeness - this is a notion that only works for metric spaces.

I propose we meet a bit earlier, around 11h20 on ZOOM to also discuss your FEEDBACK (thanks to everyone who bothered to answer!) and the EXAMS.

We will first finish linking together compactness and completeness: we prove a metric space is compact, if it is totally bounded and and complete.
Thereafter we start working with the set of continuous functions from a metric space to a metric space. This will be an occasion for us to see how everything we have learned about compactness, completeness etc 
will help us prove quite deep theorems, like Arzela-Ascoli theorem that you have used in complex analysis.

We will meet at 11h30 on ZOOM to quickly go through the QUIZ and then discuss the questions you might have.
Part of the videos are up, part will come quite a bit later today, because I just realized that the longest videos I recorded did in fact not record for technical reasons...so I have to redo them. Apologies! :)

We will prove Arzela-Ascoli theorem and look at how to talk about generic properties in topological spaces.
We meet at 11h40 in ZOOM for a Q&A session. 
The videos will be there only very late, unfortunately - apologies.

The final week of the course. Basically the material is the Baire Category theorem, but you might have already looked into it.
I have prepared some materials for revision, and a mini-mock-exam.

I propose we meet at 11h15 on ZOOM to discuss the exam, the mock-exam and the revision.