Given a set of points $S$ in some metric space, a pair of points $x, x'$ will be termed $\epsilon$-connected if they are connected by a series of points $x_1, \ldots, x_m \in S$ such that $d(x, x_1)$, $d(x_1, x_2), \ldots, d(x_{m-1}, x_m)$, $d(x_m, x')$ are all at most $\epsilon$.
Suppose $n$ points are drawn from a fixed probability distribution $\mathcal{D}$ on some metric space. I would like to say that $D$ is [SOMETHING] if for any fixed pair of points $p, p'$ and any $\epsilon>0$ as $n \to \infty$ the probability that $p$ and $p'$ will be $\epsilon$-connected goes to 1.
Is there a common term for this type of thing?