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](https://en.wikipedia.org/wiki/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 $M$ is defined independently of the metric tensor $g$ (and its signature, causal structure, curvature, etc). 

5. Also one should not conflate a [metric](https://en.wikipedia.org/wiki/Metric_(mathematics)) $d:M\times M\to [0,\infty[ $ in a [metric space](https://en.wikipedia.org/wiki/Metric_space) (within the framework of [topological spaces](https://en.wikipedia.org/wiki/Topological_space) and [general topology](https://en.wikipedia.org/wiki/General_topology)) with a metric tensor $g$.