>>11824302Consider for instance the taxicab metric on Z^2. Such a metric induces a discrete topology (in the sense that each point has a neighborhood containing only itself)
Now to answer OP's question: first of all, to the physicist, these kinds of questions are not approachable. Physicists are much more concerned with questions such as, "Which kind of (mathematical) space admits a model best aligned with experiential results?"
It turns out that the answer to this new question is surprisingly rich. My physics isn't the best but I believe discrete space breaks CPT (i.e. universal symmetry) and relativity, but does not break QFT, and in fact gives rise to quantum gravity. So, as is often the case, the answer seems to lie in the relationship between QFT and relativity. Regardless it seems the hope is that space is continuous, since this is very intuitive and does not break special relativity outright.
