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

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 $p\in M$ 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](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 $g$.

6. If we try to use a metric tensor $g$ of indefinite signature to construct a metric $d$ from [geodesic distance](https://en.wikipedia.org/wiki/Geodesics_in_general_relativity), it would for starters violate the [Hausdorff property](https://en.wikipedia.org/wiki/Hausdorff_space) and possibly the non-negativity of $d$.

7. For a [Lorentzian manifold](https://mathworld.wolfram.com/LorentzianManifold.html) $(M,g)$, 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](https://en.wikipedia.org/wiki/General_topology) $\tau$. Here $I^{\pm}(p)$ is the [chronological future/past](https://en.wikipedia.org/wiki/Causal_structure) of the point $p\in M$, respectively.