Q:

What is the topology of the Galilean and Newtonian spacetimes?

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!

A:

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.

