Coarsely dense subsets, quasi-inverses and quasi-isometries

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.

A subset $A\subseteq Y$ is coarsely dense in $Y$ if there is a real $R\ge 0$ such that for every $y\in Y$ there is an $a\in A$ with $d_Y(y,a)\le R$. Thus every point of $Y$ lies within one uniform bound of an actual point of $A$. When $A$ is nonempty this is equivalent, after enlarging the bound if necessary, to boundedness of the point-to-set distance from Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space. The quantified form also covers the empty space without invoking the undefined expression $d(y,\mathrm{\varnothing })$.

Let $f:X\to Y$ and $g:Y\to X$ be coarse Lipschitz maps. Then $g$ is a quasi-inverse of $f$ if both composites $g\circ f$ and ${\mathrm{id}}_X$, and $f\circ g$ and ${\mathrm{id}}_Y$, are at bounded distance in the sense of Bounded distance between two maps into a metric space.

A quasi-isometry is a coarse Lipschitz map that admits a coarse Lipschitz quasi-inverse. Two metric spaces are quasi-isometric if some quasi-isometry between them exists.