Choice of metric/topology on $\mathbb{R}^n$ when we say a manifold is locally homeomorphic to it

Question

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 (thanks to mike stone for this): 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.

Answer 1

1. A pseudo-Riemannian manifold $(M,g)$ of signature $(r,s)$ is a differentiable manifold $M$ of dimension $n=r+s$ equipped with a metric tensor $g\in\Gamma({\rm Sym}^2T^{\ast}M)$ of signature $(r,s)$.

2. A differentiable manifold $M$ is a topological manifold with a globally defined differential structure.

3. A topological manifold $M$ of dimension $n$ is a locally Euclidean Hausdorff space, i.e. every point $p\in M$ has a neighbourhood which is homeomorphic to $\mathbb{R}^n$.

4. Notice in particular that the underlying topological manifold $M$ is defined independently of the metric tensor $g$ (and its signature, causal structure, curvature, etc).

5. Also one should not conflate a metric $d:M\times M\to [0,\infty[ $ in a metric space (within the framework of topological spaces and general topology) with a metric tensor $g$.

6. If we try to use a metric tensor $g$ of indefinite signature to construct a metric $d$ from geodesic distance, it would for starters violate the Hausdorff property and possibly the non-negativity of $d$.

7. For a Lorentzian manifold $(M,g)$, diamond sets of the form $$ I^+(p)\cap I^-(q) , \qquad p,q\in M, $$ and their finite intersections generate all open sets $\{G\subseteq M \mid G\in\tau\}$ for the underlying locally Euclidean topology $\tau$. Here $I^{\pm}(p)$ is the chronological future/past of the point $p\in M$, respectively.

Answer 2

The topology on the mathematical model of spacetime used in general relativity is the standard topology of ${\mathbb R}^4$ that is induced from the usual Euclidean metric on ${\mathbb R}^4$. It is not a topology induced from the Minkowski metric.