Topology on flat space when a manifold is locally homeomorphic to it

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lavinia
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[I urge the viewer to read the full post before trying to reply]

I'm watching Schuller's lectures on gravitation on youtube. It's mentioned that spacetime is modelled as a topological manifold (with a bunch of additional structure that's not relevant to this question).

A topological manifold is a set ##M## with a topology ##\mathcal{O}_M## such that each point in ##M## is covered by a chart ##(U,x)##, where ##U\in\mathcal{O}_M## and ##x:U\to x(U)\subset\mathbb{R}^n## is a homeomorphism. To even talk about the map ##x## being homeomorphic, we need to be able to talk about open sets in, and hence a topology on, ##\mathbb{R}^n##.

The instructor mentions (see here) that ##\mathbb{R}^n## is considered to have standard topology. Standard topology is defined on the basis of open balls around points in ##\mathbb{R}^n##. To define open balls we need to specify a metric on ##\mathbb{R}^n##, and the definition of open balls in lecture 1 of the series was given assuming a Euclidean metric on ##\mathbb{R}^n##, i.e., $$B_r(p)=\{q\in\mathbb{R}^n\ |\ \|p-q\|_E<r\}$$ where ##\|\cdot\|_E## is the Euclidean norm.

So I wonder, is assuming Euclidean metric necessary? I've heard that curved spacetime is modeled as a manifold that locally looks like flat spacetime, which is modeled as Minkowski space as far as I know, which in turn has the Minkowski metric.

If that's the case, then charts on curved spacetime are locally homeomorphic to open sets in Minkowski space. Would we have to define the topology on ##\mathbb{R}^4## as a variant of the standard topology in which open balls are defined as per the Minkowski metric? i.e.
$$B_r(p)=\{q\in\mathbb{R}^4\ |\ \|p-q\|_M<r\}$$ where ##\|\cdot\|_M## is the Minkowski norm corresponding to metric signature ##\text{diag}(-1,1,1,1)##. I imagine this could be tricky to define since Minkowski metric isn't positive definite.

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Slightly more elaboration on my thought process: The topology is what decides the "closeness" of points in a set as far as I know. So essentially when we're approximating a small patch of curved spacetime by the flat Minkowski spacetime, if we're assuming standard topology characterized by the Euclidean metric, what we're saying is: the Euclidean metric decides the closeness of points in (locally approximated) Minkowski space.

This sounds contradictory because physical considerations scream at us that spacetime intervals (a measure of closeness of Minkowski spacetime points) are measured using the Minkowski metric.
One does not need to define the topology on Euclidean space by making it into a metric space. If one knows the topology of the real line then Euclidean space has the product topology. On the other hand, one can use the Euclidean metric or any other metric whose open balls provide a basis for its topology.

On an abstract manifold there is no canonical metric. In differential geometry a metric is an additional structure and usually means a smoothly varying non-degenerate symmetric bilinear form on the tangent spaces. If the form is positive definite them one can make the manifold into a metric space by defining the distance between two points to be the greatest lower bound of the lengths of piece-wise smooth paths connecting the two points. It is a theorem that this distance measure determines the same topology on the manifold as the topology defined through coordinate charts. If the manifold is geodesically complete - any geodesic can be extended indefinitely - then the distance between two points is the length of the shortest geodesic connecting them. This geodesic may not be unique.

BTW: Open balls are points of distance less than a given amount from a given point in a metric space. They are not all of the open sets in a topology. However every open set is a union of open balls so a test that a set is open is whether it contains an open ball around each of its points.

In the definition of a manifold, one does not require that coordinate charts be homeomorphisms with open balls in Euclidean space. One only requires the sets to be open in Euclidean space. But clearly by restriction of the chart one can choose a sub chart that does map a set homomorphically onto an open ball.

One needs to keep in mind that there are two uses of the word metric here. One is a smoothly varying symmetric bilinear form on the tangent spaces. The other is a distance measure on the manifold. A metric on the tangent spaces allows one to measure the lengths of tangent vectors and by integration, the lengths of paths. From the lengths of paths one gets a metric (measure of distance) on the manifold when the bilinear form is positive definite.
 
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