The following content has been provided by the University of Erlangen-Nürnberg.

So good morning and welcome back. Today we'll start a new chapter, namely topological manifolds and bundles.

And very likely this will be, at least in parts, new territory for you.

Because what we did so far was to revisit material that you very likely saw already in one form or the other.

But I assume the latest, starting with bundles, there will be some new results and some new type of thinking associated with this.

Topological manifolds are still topological spaces, but they're very special topological spaces.

And in fact, they're so special that nobody would use simply the same term topological space for them,

but they deserve an extra term, and that's topological manifolds.

And all the later manifolds we're going to construct, differentiable manifolds and so on, they will all still be topological manifolds.

And bundles will be a way to think about topological manifolds that have a certain structure

and that underlies much of modern physics. For instance, the standard model of particle physics, in particular, non-abelian gauge theories and so on.

In principle, you understand, or well, really understand the structure only if you understand the structure of bundles,

and introducing the language of bundles is very useful for that.

But we first start with topological manifolds, and very roughly speaking, a topological manifold

is a topological space that locally looks like some Rd.

So roughly, topological manifold is topological space, is a topological space that locally,

and that's the key word here, that locally looks like some Rd for a fixed number d.

And so it's easy to draw some. So the surface of this torus, the surface of it, that's a topological manifold, T2.

And we had the sphere, so again, the surface of the sphere, S2, that's a topological manifold, looks locally like R2, both of them.

This guy is a topological manifold, again, for d equals 2, so obviously it's particularly easy to draw examples for d equals 2,

so T2, S2, so this is the bretzel, and so on, and of course, also for higher d.

But so what precisely is a topological manifold? So precisely, we have the following definition.

A paracompact Hausdorff topological space, MO. So you see we raise paracompact and Hausdorff to conditions here.

This is called a d-dimensional topological manifold.

And usually, we do not say topological manifold, we just say manifold,

and this is used here as a qualifier to distinguish it from manifolds that have even more structure, which we're going to look at later.

It's called a d-dimensional topological manifold if for every point P in the set M, there exists a neighborhood,

there exists an open neighborhood, that means there exists a U that contains the point,

and the U is an open set of the topological space if for every point or around every point,

there exists an open neighborhood and a homeo, a homeo with its full name, a homeomorphism, let's call it X,

that takes you from this open neighborhood U into some subset of RD, so into the image, I'm sorry, into the image X of U,

which is a subset of some RD, and that must be, you see, for every point X, for every point P, this must be the same D, must be a subset of the same R to the D,

and a homeomorphism X from U into X U RD, where if we talk about a homeomorphism, we need to know the topology on this space and topology on that space in particular,

well, RD we equip with the standard topology and X of U with the induced topology, that's fine,

and the U is already an open set in MO, so the induced topology will yield what you need here.

So that's the definition, and that's this locality, so intuitively speaking, if you look at these pictures I drew,

then in there, you've got to ask yourself, is for every point of this space, is it true that around every P there exists an open set U that contains P,

well, without the boundary, if you think in standard topology terms, such that this set U is mapped by some homeomorphism into some subset of RD,

into some subset of RD, okay, and obviously, so you look at this space and say, sure, locally, if you just look very closely,

this looks like a piece of RD from the topological point of view, it can be deformed, such that it looks like a piece of RD.

Okay, so that's the definition, well, examples, I already drew some, maybe it's worth mentioning, worth emphasizing,

that looking at the whole thing through topological glasses, so looking through topological glasses, if you wish,

what does that mean, well, it means to identify all the homeomorphic manifolds, that is, identify all homeomorphic manifolds,

we see that, for instance, S2, which is homeomorphic to the set, if I think that this is embedded in R2 and it inherits the topology from R2,

the standard topology, but that's topologically the same as if you drew a square, so it doesn't know shape, right,

whereas the square can be continuously deformed into the circle and so on, so you could even, if you embed in R3,

you don't need to embed, but you could embed, and you look at this guy here, but this doesn't mean that the curve ends,

but it's supposed to have some 3D thing that this line runs behind this line from the point from where you look and so on,

so all these guys here are actually as a topological manifold are this S2 and the real line, if you wish, is topologically equivalent,

that's the whole line, but that's the same topologically as this guy, and that's the same as and so on,