My question is, given two space-time points $x^{\mu}$ and $y^{\mu}$, if the events that occur at these points are simultaneous, i.e. $x^{0}=y^{0}$, are the two events necessarily space-like separated? The reason I ask is that I'm trying to understand the notion of equal-time commutation relations in QFT (in which the commutator is non-zero in the case where $\mathbf{x}=\mathbf{y}$).
For example, if one has a field $\phi$ and its conjugate momentum $\Pi_{\phi}(y)$, then the commutation relation between them is given by $$[\phi (t,\mathbf{x}),\Pi_{\phi}(t,\mathbf{y})]=i\delta^{(3)}(\mathbf{x}-\mathbf{y})$$ Now is the reason for this being equal to a $\delta$-function because of locality? i.e. given that the two fields are evaluated at the same time, then as locality demands that they can only "communicate" if they are separated by a time-like separation, they must necessarily be evaluated at the same spatial point, as if $\mathbf{x}\neq\mathbf{y}$ then there would be a space-like separation between the two fields (as $\Delta s^{2}=(x^{0}-y^{0})^{2}-(\mathbf{x}-\mathbf{y})^{2}=-(\mathbf{x}-\mathbf{y})^{2}<0$), and they would therefore commute (in order to obey locality)?