Restriction on distance between two points in a dense subset.

Question

Consider a metric space $(M,d)$. Under what conditions is it the case that there exists some dense subset $D$ of $M$ such that $$\forall x,y \in D \, d(x,y) \neq 1$$ So far I have proven that such a dense subset exists in $\mathbb{R}^n$ by taking rationals with odd numerator and denominator, and I conjecture that such a subset exists for all second countable metric spaces.