Could we specify that a quantum space-time (any) belongs to what mathematical spaces? e.g to discrete spaces? Apparently, there is not a unique and generally accepted definition for a quantum space-time yet. In any case, I'd like to know that, at this stage, could we specify that a quantum space-time (any) belongs to what sorts of mathematical (or algebraic) spaces? For example, discrete spaces? Sorry if the question doesn't make sense somehow.
 A: The notion of a discrete space-time is a form of quantum gravity commonly known as loop quantum gravity (there are many variants on this theme with different names). Lee Smolin is one of the leading investigators in this approach, but it faces issues with Lorentz invariance violations in many formulations which is strongly disfavored experimentally. In those theories, the number of dimensions is often an "emergent" property of the deeper structure of spacetime as is the concept of locality.
Many versions of string theory require 10 or 11 dimensions of space time, usually with all but four either "compactified" or with all particles except gravitons confined to a four dimensional "brane."
The most common kinds of dimensions are continuous and smooth dimensions that map to the real numbers, those that map to the complex numbers, and finite dimensions. Various topologies of space-time are possible with those that have concave curvature, convex curvature, and torus topology making up the most notable exceptions to flat space-times. 
The usual space-time of the Standard Model of Particle Physics is called "Minkowski space"
A: The influence of mass on the space-time geometry can be described with quantum theory, where we can state the expected effect of mass on curvature of space-time.
Also it can be theorized that if mass has influence on space-time curvature, in quantum situations where mass does not curve space-time at measured coordinates, it will have done so at other coordinates. The probability of mass affecting space-time in normal situations equals the general relativity (GR).
At extreme situations such as Big Bang and black holes, the behavior of mass differs from "normal" situations. The effect of gravity, bringing particles in closer proximity, increases the chance of two particles taking up the same coordinates in space-time, which is impossible. The chance of dislocation thereby increases.
It is due to this dislocation that there is no such thing as singularity. GR calculations of minimum radius of the mass contained confirm this too. It is for this reason the big bounce theory becomes more popular again.
All matter turns into dark matter over time due to gravitational pull/Higgs field interactions, creating micro black holes throughout the universe (dark matter). These continue their existence until they clog together, bounce back etc. 
