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A pseudo-Riemannian manifold $(M,g)$ of signature $(r,s)$ is a differentiable manifold $M$ of dimension $n=r+s$ equipped with a metric tensor $g\in\Gamma({\rm Sym}^2T^{\ast}M)$ of signature $(r,s)$.

A differentiable manifold $M$ is a topological manifold with a globally defined differential structure.

A topological manifold $M$ of dimension $n$ is a locally Euclidean Hausdorff space, i.e. every point $p\in M$ has a neighbourhood which is homeomorphic to $\mathbb{R}^n$.

Notice in particular that the underlying topological manifold $M$ is defined independently of the metric tensor $g$ (and its signature, causal structure, curvature, etc).

Also one should not conflate a metric $d:M\times M\to [0,\infty[ $ in a metric space (within the framework of topological spaces and general topology) with a metric tensor $g$.

If we try to use a metric tensor $g$ of indefinite signature to construct a metric $d$ from geodesic distance, it would for starters violate the Hausdorff property and possibly the non-negativity of $d$.

For a Lorentzian manifold $(M,g)$, 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 $\tau$. Here $I^{\pm}(p)$ is the chronological future/past of the point $p\in M$, respectively.