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Now, let's start. Nevertheless, there is a whole number, well, actually an entire zoo of topological properties that are heavily used.

There is separation properties, which I'm going to write down now. Then there is connectedness, path connectedness, simple connectedness, house of separation property.

There is compactness, paracompactness, dot dot dot, it's a very long list.

There's actually a book that's, I don't know what the title is, it's kind of topological properties.

And for many, many spaces, hundreds of properties are listed and whether this space has the property or not.

But no such set is sufficient. You always find sets, two sets that are not homeomorphic, but share all those properties.

So, okay. Anyway, so, but these, apart from serving in a future classification possibly, these topological properties are important in themselves.

And so we start with topological properties, topological properties.

One, separation axioms or separation properties.

Definition. A topological space M O is called T 1.

That's the property, it's called T 1. It's called T 1 if for any two distinct points, that means P point P, that's not the same as the point Q.

There exists an open neighborhood of P, there exists a U in the topology that contains P, there exists an open neighborhood of P, such that P does not lie in U.

Or one can write this differently, that the set with the element P does not intersect U, meaning that the intersection of this one point and the U is empty.

Such a space is called T 1.

Definition. A topological space M O is called T 2, and that has a nicer name, that's called Hausdorff.

Hausdorff actually came up with the topological axioms, he formulated them in terms of neighborhoods, but they're fully equivalent to the set I wrote down, to the set of axioms I wrote down in the beginning, defining it by open sets.

But Hausdorff had one additional axiom compared to the three, and that one is called the Hausdorff axiom.

Nowadays, because so many interesting topological spaces do not have that property, one removes that axiom from the general definition of topology, but because for many spaces, I mean Hausdorff was very clever, he came up with the whole thing in this abstraction,

and you want this additional property, you call it T 2, or Hausdorff, if for any two distinct points, for any P not Q points, like here, there exists an open neighborhood of P and an open neighborhood of Q, such that the intersection of the two neighborhoods is empty.

So if one wants to draw a picture intuitively, well intuitively we better say what topology we take, we take R 2 and equip it with the standard topology, I have this point and this point, if they're distinct, I always find an open set around here,

and an open set around here, such that the open sets do not intersect, it's a separation axiom. Now this here, the T 1 property is, for any two distinct points P and Q, you always find an open set around P, such that the Q doesn't lie in there.

You see this is stronger, here you need two clouds around each point, two open neighborhoods, here you need only one such that this doesn't lie in there. And there are examples for sets that are T 1 but not T 2 and so on.

So example, R D with the standard topology on R D is T 2, and hence it's T 1, right, T 1 is weaker than T 2, but then there are other examples like for instance the Zariski topology,

that's a topology that appears in algebraic geometry, that is T 1 but not Hausdorff, but it's not T 2, and there will be a problem on the problem sheet for the Zariski topology, and the Zariski topology is very important in algebraic geometry and pushed the subject forward.

So that's a very important topology. And there are further separation axioms, so remark, there's also T 3, sorry, more impressively T 2 and a half, like this.

So T 2 and a half is, you have two distinct points, and around one of them there's actually a closed neighborhood, such that it doesn't intersect with the other point.

So that's T 2 and a half, and then there's T 3 and T 4 and whatnot. There are stronger and stronger separation axioms, and the original one by Hausdorff is that one, but you see there are situations in which you want one and then the other and so on.

So T 3, what is T 3? T 3 is you have not two points, but you have a closed set and a point, and around the closed set you find an open set such that that doesn't intersect with the point.

So you come up with crazy stuff. They all have their application in some branch of mathematics.

Also on the problem sheet will be the proof that in a Hausdorff space, so in a topological space that is Hausdorff, you can prove that every sequence that converges has a unique limit point.

That's a very intuitive notion. It converges to a point where it doesn't converge to another point.

Well, for the chaotic topology, that's not the case. As we saw, every sequence converges against any point. Well, if that is true, what I just said that in a Hausdorff space, if it converges, the convergence point is unique, that already shows that the chaotic topology cannot be a Hausdorff space.

Of course it's not. So count the example to T 2. Counter example to T 2 is M equipped with the chaotic topology because you choose any two points in M, and now you need to find, well, it's not even T 1.

It's not even T 1 because you will never find an open set in which P lies that doesn't contain Q because the only open set in which it can lie is all of M.

That's on the problem sheet. So we continue our list of topological properties, and as I mentioned, there is no complete list of topological properties that you could require for a topological space,

such that if you had two topological spaces which differed in at least one of this complete list of topological properties, you could already conclude they're not homeomorphic. Indeed, no such list is known.

Nevertheless, in many circumstances, it's very convenient to add to the axioms of a general topological space additional special topological properties, sometimes in order to be able to prove some theorems that you couldn't prove without having that property,

sometimes in order to understand the situation better. It very much depends. Now, the next pair of top properties I would like to talk about are compactness and paracompactness.

So these are two very heavily used topological properties because indeed, very often, if you want to prove some theorem in some setting to which a topological space is an underlying structure,

very often you would first tackle the proof in a compact case because usually proofs are simpler there, and once you understood the problem in a compact case, you'd try to extend the theorem to the non-compact case, which sometimes simply is not possible, but sometimes it is.

Paracompactness is a much weaker notion than compactness. In fact, it's so weak that it's very difficult to find any topological spaces that appear naturally, if you wish, that are not paracompact.

I present a theorem to that effect, and for manifold theory, which will be our concern later, paracompactness is a fundamental assumption that we will always make, similar to the Hausdorff assumption, because Hausdorff, so that was the T2 separation property,

Hausdorff plus paracompactness gives us a very convenient property of a space, namely that there exists a so-called partition of unity. That is what I'm going to present now. But let's first start with the definition of compactness.

A topological space M O is called compact if every open cover has a finite sub-cover.

What does that mean? What is an open cover to a topological space? That's a notion that's useful to know on its own.

An open cover, so if C is a subset of the topology of the space, so it's a collection of open sets, then C subset of O is called an open cover if, very simply, the union of all the sets in C already recovers the entire topological space.

It's the very intuitive notion of covering the topological space. That's an open cover. What is a finite sub-cover?

So let's be more specific. If every open cover C has a finite sub-cover C twiddle. A finite sub-cover is a cover C twiddle, so a sub-cover is not less than a cover, it's certainly as well a cover.

So there exists a sub-cover C twiddle such that for every element in the cover, for every open set in the cover we have, there exists an element U twiddle in C twiddle such that U...