About physical meaning of Hausdorff, Second Countable and Paracompact conditions of Manifold Theory

Question

I would like to ask you, specially for the people here who deals with General Relativity/Differential Geometry, physical implications about Manifolds. Well, the most intuitive notion about a manifold as a model of physical reality, is then the notion of the "space of all events". Or even, the Earth itself: localy the world a, priori, appears to be flat, but if you take a high orbital flight you will see a $S^{2}$ space.

But, concerning the mathematics of a manifold, well, we then have some ingredients [1],[2],[3]:

I - Basic Structures:

1) A Set $X$

2) A Topology in the set $X$: $\tau_{X}$

3) With 1) and 2) we can create a topological space $M \equiv (X, \tau_{X})$

4) An open Neibourhood $U_{i} \in \tau_{X} $ of a point $p \in M$

5) Chart $\phi_{i}: U_{i} \subset M \to \mathbb{R}^{n}$; $\phi$ is and homeomorphism

6) With 4),5) we have an Atlas for $M$, $\mathfrak{A} = (U_{i} ,\phi_{i})$

.

II - Manifold Theory:

7) With the itens in section I, we can then define a structure which is "localy Euclidean": the Topological Manifolds. Then a Topological Manifold is a structure that is:

i) A Topological Space with 6)

ii) $U_{i}$ must to cover $M$, i.e., $\cup _{i} U_{i} =M$

8) A Topological Manifold is a Manifold when:

i) Given $p \in U_{i}$ and $p \in U_{j}$ such that $U_{i} \cap U_{j} \neq \{\}$, then the map $\psi_{ij} =: \phi_{i} \circ \phi_{j}^{-1}$, from $\phi_{j}(U_{i} \cap U_{j})$ to $\phi_{i}(U_{i} \cap U_{j})$ is infinite differentiable

ii) The open sets of the Topological Manifold $M$, obeys the Hausdorff condition: Given $U_{i}$ and $U_{j}$, then $U_{i} \cap U_{j} = \{\}$

iii) The Topological Manifold is second countable

iv) The topological Manifold is Paracompact

My question lies about the physical motivations of a Topological Manifold be paracompact, second countable and hausdorff and I need this for a conceptual introduction for a project which I'm enrolled.

Now, the necessity of a topological space can be motivated by saying that "we have a necessity of a notion of continuity to define continuos functions and then limits, derivatives, integrals etc...", but Hausdorff, Paracompact and Second Countability are totally alien to me.

Futhermore, I would like to "see" these conditions in, for example, Newtonian Physics or even in General Relativity. What I'm trying to ask is why the Manifold Model is a good thing to describe our world, and what the world would appear if the conditions of been Hausdorff, Second Countable and Paracompact doesn't make any difference. I'm trying to understand physically these conditions, both for General Relativity and for Newtonian Physics.

$------------$

-The book, "Road to Reality" (by Penrose), doesn't help me at all.

$$ * * * $$ For construct this question and the sections $I$ and $II$ I read the definitions from 3 different sources:

[1] Introduction to Topological Manifolds (Lee); Pages $31 - 35$.

[2] Geometry, Topology and Physics (Nakahara); Pages $171 - 172$.

[3] https://en.wikipedia.org/wiki/Topological_manifold

Answer 1

One place where you may find some motivations for the general topology of a spacetime is in Geroch and Horowitz's article in "General relativity: an Einstein centenary survey". Here's a few general ideas of why spacetimes should indeed be so (those are only hints as of course there are theories where this is not the case, although for those properties it is fairly rare).

Spacetime should be connected. It doesn't have to be of course, and it's almost trivial to consider non-connected spacetimes, but if a spacetime isn't connected then neither is it path connected, and thus we can't really probe any of the components that isn't ours, since any such experiment can roughly be modelled with curves in spacetimes.

Spacetime should be Hausdorff. As much as I enjoy talking about non-Hausdorff spacetimes (there's a whole theory on them that has extremely interesting reasons behind it yet is totally not worth it), the Hausdorff property is used to have unique limits. In a non-Hausdorff spacetime, curves may split into multiple curves due to this. Non-Hausdorff manifolds do not admit a partition of unity, which is a very useful tool. Due to the splitting of curves, vector flow is not unique. As two different points can have a distance of zero, any Riemannian metric will not generate the topology of the manifold. Stokes' theorem breaks down. It's still possible to make a proper theory of spacetime with such manifolds, but things get much more complicated for something that doesn't really seem necessary. Even if we assume a branching manifold as in those theories, in any reasonable case an observer would travel along a single $H$-manifold (a subset of a non-Hausdorff manifold that is entirely Hausdorff), and it would therefore be impossible to distinguish by experiment, much like the case of connectedness.

Spacetime should be paracompact and second-countable. The fundamental reason behind this is is that we're going general relativity, and therefore we need a Lorentz metric on our manifold. A Lorentz metric requires the possibility of a Riemannian metric, which requires the manifold to be a metrizable topological space. By Smirnov's metrization theorem, a metric space must be paracompact. Any space that is both locally Euclidian and paracompact is, by theorem, second-countable.

There are plenty of non-paracompact manifolds we could take as examples, the simplest one is of course the long line $\mathbb{L}$, which is roughly obtained by taking $\omega_1$ (the first uncountable ordinal) copies of $[0, 1)$, put the basic lexicographic topology to order it, and mirroring it around zero. Locally this simply looks like $\mathbb{R}$, but it is very long (hence the name). You could construct $\mathbb{R}$ the same way but picking the countable ordinal $\omega$ instead, in which case we could take a dumb distance like

$$\forall x \in \mathbb{R}, \exists n \in \omega, y \in [0,1), x = (n, y), d(x_1,x_2) = |(n_2 + y_2) - (n_1 + y_1)|$$

This works out because the distances between points are roughly in $\mathbb{N}$. The long line, by comparison, is long. The "distance" between two points can be much larger than any natural number, and therefore there's no map to $\mathbb{R}$ between two points sufficiently far apart (although it is always locally metrizable but then again that's always true of manifolds since they are locally $\mathbb{R}^n$). If our universe is not paracompact, we can simply model it as a paracompact subset of it as any further region would be too far off to experiment on.

Answer 2

The following is a quote from Analytical Mechanics by Valter Moretti, p. 29. It explains the physical motivation for Hausdorffness.

Because of the experimental errors made by measuring instruments, any spacetime determination is only approximatively accurate. We may locate an event only with a certain approximation. In mathematical terms, we may only detect neighbourhoods of events and not the events themselves. These neighbourhoods are determined by the imprecisions, as small as we like but non-zero, of the instruments that measure space and time. That the spacetime topology is Hausdorff means that we can always distinguish two events so long as the measurements are sufficiently precise, even if not infinitely precise.

I don't know anything about general relativity, so this answer deals with Newtonian physics only.