Is it possible to model spacetime as a function from an interval $I$ to a set $S$, where $t\in I$ represents time and $x\in S$ represents space?

Question

According to wikipedia, spacetime is modeled via a pseudo-Riemannian 4-manifold, with one dimension representing time and the other three dimensions representing space. I would like to know whether that manifold can possibly be viewed as a (surjective) function $F$ from a real interval $I$ onto a set $S$, where each point $t\in I$ represents time and each $F(t)\in S$ represents a collection of points in space (e.g. $F(t)$ represents all of space at time $t$).

Obviously this is not always possible for Riemannian 4-manifolds, e.g. the unit sphere in $\mathbb{R}^5$. But I don't need it to always be possible. On the other hand, I do need something that comports with physics; so I can't just take $\mathbb{R}^4$ itself, for instance.

Any help would be appreciated.

A function $f: I \to S$ is a curve in space. Do you mean a set of the form $I \times S$? And why can't you take $\mathbb{R}^4$? Your question is not very clear to me. — Javier, Mar 05 '20 at 14:13
Related: What is the closest general-relativistic equivalent of a “time slice”? — safesphere, Mar 05 '20 at 15:15
It is unclear if your time interval is continuous. For example, in the Schwarzschild coordinates, time first goes to infinity, but then continues from the Schwarzschild radius to zero. It is also unclear if your question is limited to physically realistic cases or includes non-physical solutions like Schwarzschild with a timelike radius or Kerr with timelike loops inside the Cauchy horizon. — safesphere, Mar 05 '20 at 16:08

Is it possible to model spacetime as a function from an interval $I$ to a set $S$, where $t\in I$ represents time and $x\in S$ represents space?

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