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Today we'll start having a fairly good look at topological spaces based on our previous developments of set theory.

And as I announced last time, topological spaces or establishing a so-called topology on the space, I'll define in a second what that is,

presents the weakest structure you can establish on a set in order to have the two very important notions of convergence and of continuity.

Convergence of sequences, of points on a set, and continuity of maps between two topological spaces.

Now, the definition I'm about to give is at first sight rather abstract on the downside. On the upside it's extremely simple.

And this definition is the result of a historical development and I'll comment occasionally on that.

So just take the definition as the simplest definition that after all mathematicians found to be of use in order to define topologies.

So the definition of a topological space is as follows.

Definition. Let M be some set. And we know what a set is from axiomatic set theory.

Then a choice. So you see you have a choice here. Then a choice and this choice is called curly O.

What is this O? It's a choice, it's a subset of the power set of M. It could be the entire power set.

So a choice of subset O of the power set of M is called a topology on M.

So this choice is called a topology on M if and now three axioms or three conditions need to be satisfied.

The first condition is that you're not entirely free in your choice. That's the idea of these three axioms.

And one condition is that certainly the empty set, which is an element of the power set of any set, must be in your pick of topology.

And likewise, the entire set must also be part of your choice. If you don't do that, you already have no topology chosen.

The second condition is if you have two sets U and V that are already in your choice.

You say, I want this one and that one. Well, if you did that, then it's a requirement that their intersection is also in your choice.

And you know that we have this funny way of writing unions and intersections like the intersection.

And then you need to provide an entire set that contains the sets you want to intersect.

So anyway, I first finished the statement. The intersection of these two sets is then required to be also part of your choice.

You cannot choose U and V, but not the intersection. You have to choose that too.

And in kindergarten, you wrote this U intersected V. And of course, if you wish, you can still define the intersection of, say, two sets here this way.

But from the last problem sheet, we established this intersection in similar fashion to the union symbol.

And the idea here is that you can only intersect, you could intersect infinitely many sets, not a question.

But you cannot intersect more sets than fit into a set.

Anyway, that's this here. And the third condition is if you have a collection of open sets.

So here I took two, but now I take an arbitrary collection of open sets.

Then the requirement is that the union of this arbitrary collection, not two, not finite, arbitrary collection, that this union again lie in your pick of sets.

We could have written this more in a more parallel fashion if I had written curly brackets around UV.

That would then be a subset of O. If two lie in there, they build a subset of this set.

Anyway, right. So if these three conditions are met, then the pick you made, the choice you made is called a topology.

That's it. That's all. Now, before I elaborate on that more, let me make an important remark.

Unless the set M only contains one element, there are many different, or there are different, usually many different, topologies O.

One can choose on one in the same set. So it's a real choice.

And we can have a little table. So if the cardinality of the set M, so this is for finite sets, is one, then how many topologies can you have?

That's the number of topologies one can have.

Well, if you have one, you can only have one topology because you then always have to take the set M itself. It consists of just one element. That's this.

You take the empty set and there are no other choices you can make.

But already if you have, if the set M has two elements, there are four different choices for the topology you could make.

If the set M has three, then there are 29 choices for the topologies you can make.

If M has four, you have 355 choices. If M is five, you have 6,942 choices.

If M is six, you have 29527 choices, so 209,527 choices. And if the set M has seven elements, you have 9,535,241 choices.

And you see already for a seven element set, it's an extreme number.

And obviously if you have an infinite set, there's an infinite number of choices for the topologies you could establish on that set.

So it's a real choice you have. And depending on what topology you have, your notion of continuity and convergence, continuity of maps and convergence of sequences,

which we're going to define solely in terms of the topology, will of course change. I'll present examples for that.

Now, however, let's first start with basic examples for topology to get familiar with the concept, to become familiar.

Well, let M be any set. On any set, we can choose the topology to consist of just the empty set and the set itself.