  In this paper we introduce a notion of the Gromov-Hausdorff distance with
boundary, denoted by $d_{GHB}$, to construct a framework of convergence of
noncomplete metric spaces. We show that a class of bounded $A$-uniform spaces
with diameter bounded from below is a complete metric space with respect to
$d_{GHB}$. As an application we show the stability of Gromov hyperbolicity,
roughly starlike property, uniformization, quasihyperbolization, and boundary
of Gromov hyperbolic spaces under appropriate notions of convergence and
assumptions.
