What is the "full" topological requirements for a (classical) spacetime?

Question

The model for (classical) spacetimes are, fundamentally, topological Manifolds. I know that this isn't the complete structure (because in fact we need to realize what is lorentz manifolds and so on...). But, the thing is, topological manifolds are the cement of spacetime definition and there's a lot of equivalent ways to use the "requirements" (paracompactness, hausdorff condition, etc...) and define properly what kind of structure we need to impose on they. The fact is that I wish to know all the "hierarchical path" to spacetime manifolds. I will explain.

The requirements are then:

1. $(\mathcal{M}, \tau)$, a topological space, of course.

Then, the next structures are listed in the following, but I don't know, properly, how is the hierachy between then (which one implies the other and so on...).

2. First Countable

3. Second-Countable

4. Connected

5. Path-Connected

6. Separable

7. Hausdorff

8. Compact

9. Paracompact

10. Metrizable

11. Metric Space

I know that we use all of these to define, properly, a spacetime, but I don't know what is the hierarchy between them. Please fell free to be redundant, because the kind of answer that I'm looking for is something like:

A spacetime $\mathfrak{M}$ is a topological manifold $(\mathcal{M}, \tau)$, which is compact, paracompact, metrizable and so on....

So, how can I use the requirements listed above, from $1$ to $11$, to bake the "full" definition of a spacetime?

$$ * * * $$

Also there's another concept that I don't know how to fit in between the requirements $1$ to $10$

1. Normal (or Regular)

Answer 1

Here is what you want for a space-time:

1. Hausdorff (so that you have uniqueness of limits of sequences and of functions with values in your manifold), paracompact (so that you can have a partition of unity, without which much of the global analysis will be impossible).

2. Locally homeomorphic to ${\mathbb R}^n$ (most of the time, $n=4$). However, in fact, you will need more, see Part 4 below. Part 2 will imply 1st countable.

1 + 2 imply metrizable, which is stronger than normal and the latter is stronger than regular. Conversely, every metrizable space which is locally homeomorphic to ${\mathbb R}^n$, is paracompact and Hausdorff.

3. Frequently, you'd want your space-time to be connected (same as path-connected, subject to Assumption 2). Most of the time, you will not want to assume compactness. Compactness would exclude such basic examples as ${\mathbb R}^{3,1}$ and products of a 3-dimensional manifold with ${\mathbb R}$.

1+2+3 will imply 2nd countable and separable. Sometimes, connected is too much, you would want at most countably many connected components. (Connected means exactly one connected component.) 1+2+2nd countable will imply paracompact with at most countably many connected components.

This takes care of almost everything on your list except for "metric." Metric (in topology) means a "distance function" $d(\cdot, \cdot)$ consistent with topology:

$\lim_{i\to\infty} p_i=p$ for a sequence in your manifold if and only if $\lim_{i\to\infty} d(p_i,p)=0$.

Most of the time, you do not want to fix such distance function in advance, you just want to know that it exists (which means "metrizable").

Definition. A topological manifold is a topological space satisfying 1 and 2 (some form of 3 is optional, for some arguments, you want 2nd countable, for other arguments, paracompact is good enough). Personally, I prefer to assume 2nd countable.

4. However, this is not all: You want to be able to "do geometry and analysis" on your manifold, which means that you want more than just a topological manifold defined above. In GR you want to be able to talk about a Lorentzian (signature $(n-1,1)$) metric (not a distance function!) on your manifold. For that, you have to require your manifold to be given a smooth structure (an atlas with smooth transition maps). You can think of such a structure as a strengthening of item 2 above. A topological manifold together with a chosen smooth structure is called a smooth manifold. This is what you really want for your space-time.

Remark. You also want to have a smooth structure if you are doing QFT: You want to be able to work with bundles, connections, curvature, etc, which will require some degree of smoothness.

Not every smooth manifold admits a Lorentzian metric (for instance, the 4-dimensional sphere does not). The necessary and sufficient condition for the existence of a Lorentzian metric on a smooth manifold $M$ is:

Every component of $M$ is either noncompact or has zero Euler characteristic.

Now, one can define a (classical) spacetime as:

A smooth 4-dimensional manifold $M$ equipped with a Lorentzian metric (i.e. a semi-Riemannian metric of signature $(3,1)$).

Since you do not mind a redundant answer: Every such $M$ will be Hausdorff, regular, normal, metrizable, separable, 1st countable, paracompact, 2nd countable, have at most countably many (path) connected components. However, in general, it will be neither connected nor compact.