The definition of the Galilean spacetime is a tuple $(\mathbb{R}^4,t_{ab},h^{ab},\nabla)$ where $t_{ab}$ (temporal metric) and $h^{ab}$ (spatial metric) are tensor fields and $\nabla$ is the coordinate derivative operator specifying the geodesic trajectories (see: Spacetime Structure).
A single metric does not work, because the speed of light is infinite, so time and space should be treated separately with the temporal metric:
$$t_{ab}=(\text{d}_a t)(\text{d}_b t)$$
and the spatial metric:
$$h^{ab}=\left(\dfrac{\partial}{\partial x}\right)^a\left(\dfrac{\partial}{\partial x}\right)^b+
\left(\dfrac{\partial}{\partial y}\right)^a\left(\dfrac{\partial}{\partial y}\right)^b+
\left(\dfrac{\partial}{\partial z}\right)^a\left(\dfrac{\partial}{\partial z}\right)^b$$
Finally, $\nabla$ on $\mathbb{R}^4$
is a unique flat derivative operator that for each coordinate $x^i$ satisfies:
$$\nabla_a\left(\dfrac{\partial}{\partial x^i}\right)^b=\mathbf{0}$$
In turn, the Newtonian spacetime is the same tuple with an additional structure $(\mathbb{R}^4,t_{ab},h^{ab},\nabla,\lambda^a)$ where $\lambda^a$ is a field that adds the preferred frame of rest:
$$\lambda^a=\left(\dfrac{\partial}{\partial t}\right)^a$$
What is the topology of the Galilean and Newtonian spacetimes?
To clarify my question on an example, the Schwarzschild spacetime is defined by the following metric:
$$ {ds}^{2} = -\left(1 – \frac{r_\mathrm{s}}{r} \right) \,dt^2 + \left(1-\frac{r_\mathrm{s}}{r}\right)^{-1} \,dr^2 + r^2 d\Theta^2 $$
Where $d\Theta^2$ denotes the spherical metric induced by the Euclidean on a two sphere, i.e.
$$ d\Theta^2 = d\theta^2 + \sin^2\theta \, d\varphi^2\;\;\; \text{and} \;\;\; r^2=\sum_{i=1}^3 dx_i^2 $$
According to What is the topology of a Schwarzschild black hole? – the topology of the Schwarzschild spacetime is $\Bbb R^2\times\Bbb S^2$, which is homeomorphic to $\Bbb R^4$ with the $r=0$ line removed (see: Is $\Bbb R^2\times\Bbb S^2$ homeomorphic to $\Bbb R^4$ with a line removed?).
Once again, the Schwarzschild spacetime is just an example to clarify the question, not a part of the question. Thank you!
Best Answer
Well, you can give it whatever topology you want, but the most obvious and natural choice would just be the ordinary Euclidean topology of $\mathbb{R}^4$. All the extra structure you are adding can be considered naturally as just extra structure on top of the usual smooth manifold structure of $\mathbb{R}^4$, which has the usual Euclidean topology.