The n-Category Café

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.

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

Back up!

Jolly good!

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!