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

This post imported from StackExchange Physics at 2019-07-09 21:55 (UTC), posted by SE-user M.N.Raia