Distinction between different points at Big Bang Singularity

As per the Big Bang model of Cosmology roughly 13.8 billion years ago a singularity exploded exponentially to eventually become the present universe.

At the present time (basically current time-slice) we have all the space which according to the said model was crunched into a singularity. My question is whether two distinct space points in the current time-slice correspond to two distinct points inside the singularity (at big bang or as soon as the notion of space-time could be considered) or could they have been a single point as well?

I haven’t studied cosmology (as evidenced by this rather immature question) so it would be really helpful to put things in a simple manner.