While the study of Manifolds are quite interresting by itself, we know that General Relativity have on it's core the study of Manifolds. When we transport the Manifold mathematical structure to General Relativity it's quite common to don't pay atention about the physical motivations about the very basic constructs, i.e.,
A Spacetime is a four-dimentional, paracompact,connected,secound countable,hausdorff, locally Euclidean, topological manifold. $\tag{1}$
I found a particular physical motivation on $[1]$ which states:
The requirement that a spacetime is paracompact, ensures that it always admits a metric with lorentzian signature $\tag{2}$
Well, this have a form of a Theorem, and better: a theorem with physical implications about a highly abstract concept (paracompactness). I mean, the need of a topological space is to stablish the physical fact of a countinuous world. I think you could say that paracompactness is a requirement because physically our classical world have lorentzian signature.
So my doubt is: how can I prove the steatament $(2)$? Or, in other words, is there some reference which have this proof already done or, could you give me a hint?
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$[1]$ DE FELICE.F; CLARKE C.J.S. Relativity on Curved Manifolds
This is false. For instance, $S^4$ satisfies all the conditions of (1) but does not admit a Lorentzian metric. (This follows from the fact that the tangent bundle of $S^4$ does not contain a line bundle as a subbundle; see the answers to this question on MO for more details and discussion.)
I would also remark that (1) is generally very poorly written with significant redundancy among the listed conditions, and gives the impression of being written by someone who is not very familiar with the topic. Definitions of "topological manifold" vary, but all of them include "locally Euclidean", and most include "Hausdorff" and either "second-countable" or "paracompact". Any second-countable Hausdorff locally Euclidean space is automatically paracompact. Any connected paracompact Hausdorff locally Euclidean space is automatically second-countable.