Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
Let's call this map $\phi: (M_1, \mathcal{M}_1) \rightarrow (M_2, \mathcal{M}_2)$.
The Lorentzian metrics $g_1$ and $g_2$ induce binary relations $\ll_1$ and $\ll_2$ over their manifolds respectively which is defined as such:
$p \ll q \leftrightarrow$ there exists a time-like path connecting $p$ to $q$
Suppose the homeomorphism $\phi$ is an $\ll$ isomorphism, meaning:
$$\forall p,q \in M_1 : p \ll_1 q \leftrightarrow \phi(p) \ll_2 \phi(q) $$
The path topology on a Lorentzian manifold is defined as such:
The finest topology over $M$ such that its open sets induce the same topology on each of the timelike curves in $M$(subspace topology), as the manifold topology does.
The question is if $\phi$ as a $\ll$-isomorphism and manifold topology homeomorphism, imply path topology homeomorphism(even if not under exactly the same map though this is less appreciated but still valuable).
