Idea behind topological manifold definition.

[center o bass]

The usual definition of an n-dimensional topological manifold M is a topological space which is 'locally Euclidean', by which we mean that:

(1) every point in M is contained in an open set which is homeomorphic to $\mathbb{R}^n$.
(2) M is second countable.
(3) M is an Hausdorff space.

Could someone explain what the idea is behind (2) and (3)? Normally I would think that locally Euclidean would refer to R^n with it's usual topology (the one induced by the Euclidean metric), but the definition above seems to be more general. It might even not be metrizable. According to http://en.wikipedia.org/wiki/Metrization_theorem one does also require the space to be 'regular'. Is perhaps regularity implied by (1)?

 

[jgens]

The locally Euclidean piece of the definition is contained entirely in point (1). The other two points just eliminate some of the pathology that can occur otherwise. The second countable restriction prevents the spaces from getting too big in a certain sense and the Hausdorff property is something that really ought to be satisfied if manifolds are to conform to our intuition. Those three points together are sufficient to guarantee the metrizability of the manifold.

 

[Office_Shredder]

The idea behind 2 is that we can imagine the following "manifold": Choose an uncountable ordinal number $\alpha$. Take $\alpha$ many copies of [0,1) and glue them together by attaching endpoints of subsequent intervals in the ordering. This gives a set which is locally a one dimensional Euclidean space, is Hausdorff, but fails hard to be second countable, since you explicitly need at least one open set in each copy of [0,1) to get a base (and therefore need uncountably many open sets to make a base).

This is called the long line/long ray/long something (depending on who you ask). It is something we don't want to count as a manifold since it violates things that we want manifolds to satisfy such as being embeddable in Euclidean space, and having sequential compactness being equivalent to compactness.

You can see more about it on wikipedia here:
http://en.wikipedia.org/wiki/Long_line_(topology )

Point (3) is basically a way of dictating that the charts glue together in a way that is nice. For example, consider the set $$(-1,1) \cup \{a \}$$
where a is an arbitrary element, that has an atlas of two charts. One is the expected map from (-1,1) to (-1,1) in R, and the other is the map on
$$(-1,0) \cup \{a \} \cup (0,1)$$
which maps a to 0 and the rest of the interval to what you would expect. The manifold has the coarsest topology possible to satisfy these being homeomorphisms. This is a second countable manfold that is locally Euclidean, and is often called the line with two origins. Notice every neighborhood of 0 intersects with a neighborhood of a and vice versa (since they includes points in (-1,1) that are nonzero). You can construct lots of non-Hausdorff examples by gluing charts together in awkward ways. The main point is that I only know that two points contained in the same chart are Hausdorff; if I have points in different charts I want to know that those points are separated in the topology as well.

 

[WannabeNewton]

(2) M is second countable.

We need partitions of unity for a number of very important existence theorems e.g. the existence of an affine connection on manifolds, the existence of a Riemannian metric on manifolds etc.

EDIT: since you have interest in general relativity, I would recommend taking a look through the following paper if you can get free access to it:

R. Geroch. Spacetime structure from a global viewpoint. In R. K. Sachs, editor, General Relativity
and Cosmology. Academic Press, 1971.

 

[blue_raver22]

The idea behind the second and third conditions in the definition of a topological manifold is to ensure that the space is well-behaved and has certain desirable properties that are necessary for studying it as a manifold.

The second condition, known as second countability, means that the space has a countable basis for its topology. This ensures that the space is not too large or complicated, and allows us to work with it in a manageable way. It also guarantees that the space is separable, meaning that it has a dense subset, which is important for many topological and analytical arguments.

The third condition, known as Hausdorffness, is a stronger separation axiom than the one implied by (1). It means that every two distinct points in the space have disjoint open neighborhoods. This ensures that the space is well-separated and avoids certain pathologies that can arise in non-Hausdorff spaces. In particular, it guarantees that limits of sequences are unique, which is important for many analytical and topological arguments.

The requirement for regularity, as mentioned in the Wikipedia article, is indeed implied by the first condition in the definition. This is because the homeomorphism between the local open set and $\mathbb{R}^n$ ensures that the space is locally compact and Hausdorff, which together imply regularity.

In summary, the idea behind the second and third conditions in the definition of a topological manifold is to ensure that the space is well-behaved and has desirable properties for studying it as a manifold. These conditions may seem technical, but they are necessary for the space to have the necessary structure for studying it using the tools of topology and analysis.