I think you could just take an open interval in the order topology and then create a collection by turning the first dimension into a parameter. IIANM for each value of the parameter you'd get an open set, they'd be pairwise disjoint, and there'd be uncountably many of them.

I didn't mean IR^n with its usual topology. I meant IR^n with the order topology for the dictionary order. IIANM you can construct an uncountable set of pairwise disjoint open intervals in this topology so it can't have a countable dense subset. But as I said it's been years since I touched a topology book.

IR as in the real numbers in fake blackboard bold :)

it's been more than a decade since i last thought about such things but would IR^n with the dictionary order work

do you mean something like this: https://www.math.colostate.edu/~hulpke/talks/decomp.pdf

commenting on the responses more than the original question:

imho taking a wait & see approach with meta is like pursuing a wait & see approach with the plague.