A metric space with sufficient points(dense enough)

Question

I say that a metric space $X$ is dense enough if $\forall r_1,r_2 \in \mathbb{R}, \forall x\in X, r_1<r_2 \Rightarrow B(x,r_1)\subsetneqq B(x,r_2)$. That is, $B(x,r)$ is strictly increasing with $r$ for any point $x$ in metric space $X$.

Questions

1. Is there a formal name for dense enough?
2. Is it true that all dense enough metric spaces are complete?
3. Are there any article or literature about such metric space?
4. Can a dense enough metric space be embeded canonically (perhaps isometry) into a Banach space?