1. A [pseudo-Riemannian manifold](https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold) $(M,g)$ of signature $(p,q)$ is a differentiable manifold $M$ of dimension $n=p+q$ equipped with a [metric tensor](https://en.wikipedia.org/wiki/Metric_tensor) $g\in\Gamma({\rm Sym}^2T^{\ast}M)$ of signature $(p,q)$.

2. A [differentiable manifold](https://en.wikipedia.org/wiki/Differentiable_manifold) $M$ is a topological manifold with a globally defined differential structure.

3. A [topological manifold](https://en.wikipedia.org/wiki/Topological_manifold) $M$ of dimension $n$ is a locally Euclidean Hausdorff space, i.e. every point in $M$ has a neighbourhood which is homeomorphic to $\mathbb{R}^n$.

4. Notice in particular that the underlying topological manifold $M$ doesn't know about the metric $g$ nor its signature, causal structure, curvature, etc.