## November 19, 2014

### General Topology Notes

#### Posted by Tom Leinster

This semester, I’ve had the pleasure of teaching a 4th year undergraduate course on topology. One of the great things about it has been the students, who are a really engaged and gifted bunch. I’ve possibly enjoyed teaching this course more than any I’ve taught before.

Another pleasure is that, having nearly reached the end of term, I find that I’ve written a complete set of notes for the course. (There’s also an introductory lecture and a set of problem sheets.) Comments welcome! It’s the first year I’ve taught this, so presumably I’ll be using some version of these notes for the next couple of years.

The course begins with the definition of topological space and takes it from there, going through standard constructions and then compactness and connectedness. It seems to me that Year 4 is pretty late to be teaching this stuff (though Scottish university students typically start a year younger than those in England or Wales, so it’s roughly equivalent to 3rd year south of the border). But on the plus side, the students had already done a good amount on metric spaces, including compactness and connectedness, and that softened up the ground nicely.

Posted at November 19, 2014 10:45 AM UTC

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### Re: General Topology Notes

I can’t help but notice that the parts of Definition A2.1 have names that look like the names of separation axioms.

Whether the separation axioms should have names T$_k$ is of course debatable (they’re mostly indexed by lifting problems which of course need not be totally ordered at all — there was a paper on that arXiv’d sometime this year, but I can’t find it now).

Posted by: Jesse C. McKeown on November 20, 2014 12:06 AM | Permalink | Reply to this

### Re: General Topology Notes

Ha! Good point.

For anyone too lazy to look up what Jesse’s talking about, I called the three axioms for a topological space “T1”, “T2” and “T3”.

When I typed that bit, I think I was imagining using those names a lot more than I eventually did. As far as I can see, I don’t even make it to the next page before giving up on them. Next year I should just use (i), (ii) and (iii).

I do, however, use the standard name $T_1$ for “points are closed” (Definition A3.6). Those $T_n$ names are pretty bonkers in my opinion. Mostly I’ve avoided them by instead saying “Hausdorff” and, on the odd occasion where they arise, “regular” and “normal”. But is there a synonym for “$T_1$”?

Posted by: Tom Leinster on November 20, 2014 1:30 AM | Permalink | Reply to this

### Re: General Topology Notes

Wikipedia says “accessible” or “Fréchet”. These names attached to people are a bit risky. If I said “Take a Urysohn/Kolmogorov/Fréchet space”, would you know I was talking about $T_{2\frac{1}{2}}$, $T_0$ or $T_1$ spaces? Hausdorff is completely standard, but the other names make it hard to tell which one is stronger than others.

Posted by: David Roberts on November 20, 2014 7:03 AM | Permalink | Reply to this

### Re: General Topology Notes

If I said “Take a Urysohn/Kolmogorov/Fréchet space”, would you know I was talking about $T_{2\frac{1}{2}}$, $T_0$ or $T_1$ spaces?

In the first case, I don’t know what either a Urysohn or $T_{2\frac{1}{2}}$ space is, so it would make no difference. In the second, I’ve known for a couple of years that “Kolmogorov” is a synonym for $T_0$, but I think the latter is much more common. In the third, no: “Fréchet” is a new one on me (thanks).

But you’re right, I’m not doing the students any favours if I use names that hardly anyone else uses.

As I understand it, the point of the $T_n$ notation is that it’s meant to be monotone: if $m \leq n$ then $T_n \implies T_m$. But in order to make that true, you have to force it. For instance, “regular” and “normal” (as defined in A3.12) are useful concepts, and are roughly equal to $T_3$ and $T_4$. But it’s not true that normal implies regular or that regular implies Hausdorff. So to get $T_4 \implies T_3 \implies T_2$, you have to define $T_3$ and $T_4$ as “regular plus Hausdorff” and “normal plus Hausdorff”.

As Jesse alluded to, the separation axioms that seem to be useful simply aren’t totally ordered. Sobriety is another example of this.

Posted by: Tom Leinster on November 20, 2014 8:23 AM | Permalink | Reply to this

### Re: General Topology Notes

If I heard you mention a “Fréchet space”, I’d assume you were talking about something else.

Posted by: Mark Meckes on November 21, 2014 8:49 PM | Permalink | Reply to this

### Re: General Topology Notes

Indeed, the shortest way to introduce the separation axioms is probably via the lifting properties wrt maps between finite spaces, as spelled out in these two papers. The first one is about the lifting property, and the other one tries to view basic topology as diagram chasing computations with preorders (but it’s not well-written and/or finished). Probably these are the arxiv papers you refer to.

So that’s a way to talk about basic topology in the category theory language.

Posted by: m on December 19, 2014 1:39 AM | Permalink | Reply to this

### Re: General Topology Notes

I get a “403 Forbidden” error when I click on the links to the notes.

Posted by: Simon Willerton on January 5, 2015 6:56 PM | Permalink | Reply to this

### Re: General Topology Notes

Yeah, sorry. For some reason yet to be uncovered, my whole web space went down over Christmas and is still not back up. My colleagues’ pages work, just not mine!

I can of course email you the notes if you want them — just let me know.

Posted by: Tom Leinster on January 5, 2015 8:39 PM | Permalink | Reply to this

### Re: General Topology Notes

No rush, I was just having a peek. Just say when they’re back up!

Posted by: Simon Willerton on January 5, 2015 10:15 PM | Permalink | Reply to this

### Re: General Topology Notes

Back up!

Posted by: Tom Leinster on January 7, 2015 4:53 PM | Permalink | Reply to this

### Re: General Topology Notes

Jolly good!

Posted by: Simon Willerton on January 8, 2015 9:35 AM | Permalink | Reply to this

### Re: General Topology Notes

In case anybody is looking for a complementary set of notes, here are notes from a General Topology course (probably Introduction to Topology would be a better title!) I taught in the spring of 2013 and 2014. They are not absolutely complete, but cover a large proportion of the course. The file is a collation of three files:

1) The first 14 lectures, up to pg.314, are on point set topology. The choice of material is very similar to Tom’s, except that I mention metric spaces only in passing (if I were to teach the course again and thereby have time to work more on the notes, more ‘extra’ sections on metric spaces would be added). A couple of lectures proving that locally compact topological spaces are exponentiable have not been written up, but otherwise the notes are complete. These notes are from 2014, a thorough revision of the 2013 notes.

2) An introduction to knot theory. These notes are from 2013 (I did not have time to revise them in 2014, but am happy with them as they are, except that I would like to Tex the pictures). The first of these lectures is handwritten (I would remove some formalities in a revision), the rest are typed.

3) An introduction to $\Delta$-sets, Euler characteristic, and the classification of surfaces. These are also from 2013, and are unfortunately not complete, missing the details of the surgery part of the argument (I did not have time to add to/revise them in 2014). They are complete as far as they go, though.

The source code is publically available if anybody is interested.

Posted by: Richard Williamson on January 8, 2015 10:53 AM | Permalink | Reply to this

### Re: General Topology Notes

Wow, that looks like nice! What level were the students, and how did it go down?

Posted by: Tom Leinster on January 8, 2015 10:52 PM | Permalink | Reply to this

### Re: General Topology Notes

My feelings when teaching the course were very similar to yours! I found it most enjoyable and rewarding, and had a very talented and engaged group of students both years.

The course is a first course in topology, taken most commonly in the second or third year by the students on the mathematics or physics degree, sometimes a little later by those on the teacher education degree (most of the students follow one of these three degree programmes) . Some had seen a very little on metric spaces before, but many had not.

I made some quite radical changes from the way the course had been taught before, in particular taking a very geometric point of view, and my impression is that the students really enjoyed and responded well to this, and appreciated what I was trying to do. Quite a few students who took the course might have been considered ‘weak’ by those taking a superficial glance, and that they thrived with the geometric emphasis of the course was wonderful to see. The feedback on the course and the notes was very positive.

The course also led to some new research directions for me, as several of the students wished to go deeper into geometric topology, and write project and master theses. The first of my master students, Therese Mardal Hagland, has just finished, and I can take the opportunity to announce that she has come up with a ‘polynomial’ (actually what I consider to be a kind of ‘2-polynomial’) invariant of 2-knots akin to the Jones polynomial for knots, constructed geometrically in a manner akin to the Kauffman bracket polynomial construction of the Jones polynomial. I am very excited about this!

Posted by: Richard Williamson on January 9, 2015 7:22 PM | Permalink | Reply to this

### Re: General Topology Notes

Years ago in Bangor we had very much the same experience with the Knots and Surfaces type of material. even apparently weak students related well to it and could follow (and use) the arguments quite well. That course did not do as much general topology as Richard’s, as we added in some combinatorial group theory so as to handle knot groups. Nick Gilbert taught the course with me and he introduced the Kauffman polynomial into it (and together we authored a textbook (OUP) based on the material). The geometric topology material provided the examples for subjects such as quotient topologies which can otherwise seem very abstract.

Posted by: Tim Porter on January 10, 2015 6:36 AM | Permalink | Reply to this

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