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My question is quite rapid. I know that the right thing to do is dig into books, but I don't have much time to spend on a proper mathematical reading.

So, considering the standard Differential Geometry and Topology, aiming to comprehend the mathematical structure of General Relativity I would like to ask:

The conditions: Hausdorff, Second-Countability and Paracompactness are required to bake a metrizable manifold? In other words, without these three ingredients I won't be able to make a metrizable manifold and therefore won't be able to define a metric tensor field?

M.N.Raia
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A manifold (locally Euclidean) topological space is metrisable iff it is Hausdorff and paracompact. Necessity is obvious, sufficiency follows from Smirnov's metrisation theorem.

Second countability is not needed (for metrisability). If you want to do analysis it might come in handy though. If a manifold is Hausdorff and second countable it will be metrisable, but the condition is not necessary.

If you want manifolds to be connected there is a theorem that a connected paracompact Hausdorff manifold is second countable (and $\sigma$-compact) so that explains why many texts focus on the second countable case.

Henno Brandsma
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