The definition of Cauchy surface in a globally hyperbolic spacetime $(M,g)$ is a set $S$ which
- is achronal
- it intersects every inextendible smooth timelike curve.
REMARKS
(1) A future-directed smooth causal curve $\gamma: (a,b) \to M$, with $a,b \in [-\infty,+\infty]$ is future, resp. past, inextendible if there is no $p \in M$ such that $\lim_{s\to b} \gamma(s) =p$, respectively, $\lim_{s\to a} \gamma(s) =p$.
(2) As $S$ is achronal it intersects every inextendible smooth timelike curve exactly once. Indeed, achronal means that there are no distinct points on $S$ which can be joined by a smooth timelike curve.
(3) Instead, acausal is a stronger conditiom: it means that there are no distinct points on $S$ which can be joined by a smooth causal curve. In other words the points on an acausal $S$ are spacelike separated. $\blacksquare$
It is possible to prove that for a Cauchy surface $S$ as defined above,
(1) $S$ is closed,
(2) $S$ must also intersect every smooth causal curve (not necessarily once however, see below),
(3) $S$ is a $C^0$ embedded submanifold of the spacetime.
An example of Cauchy surface is the plane $x^0=x^1$ in 4D Minkowski spacetime. It is clear that this is achronal but not acausal. In other words, the points on $S$ are not spacelike separated, and this answers your question.
There is a type of Cauchy surfaces which also are acausal, they are smooth spacelike Cauchy surfaces.
An important result by Bernal and Sanchez establishes the existence of these types of Cauchy surfaces when a generic Cauchy surface exists.
Presumably you mean this type of Cauchy surfaces when you speak of hypersurface of simultaneity. Tha is beacuse it is possible to prove that given such a Cauchy surface $S$, there is a (surjective) smooth function $t: M \to \mathbb{R}$ whose differential is everywhere timelike and past directed, such that a level set of $t$ is exactly $S$. That is a so-called temporal function and, using it, the spacetime turns out to be diffeomorphic to $\mathbb{R} \times S$ with metric accordingly decomposed $-dt^2 + h(t)$.
If $S$ is a generic Cauchy surface several weaker theorems are however valid (see the references below).
Some modern important references on these issues are listed below.
Antonio N. Bernal, Miguel Sanchez
On Smooth Cauchy Hypersurfaces
and Geroch’s Splitting Theorem Commun. Math. Phys. 243, 461–470 (2003)
Antonio N. Bernal, Miguel Sanchez
Smoothness of Time Functions and the Metric Splitting
of Globally Hyperbolic Spacetimes
Commun. Math. Phys. 257, 43–50 (2005)
Antonio N. Bernal, Miguel Sanchez
Further Results on the Smoothability of Cauchy
Hypersurfaces and Cauchy Time Functions
Letters in Mathematical Physics (2006) 77:183–197
