  Let $\Omega$ be a class of unital $\rm C^{*}$-algebras. The class of ${\rm
C^*}$-algebras which are asymptotical tracially in $\Omega$, denoted by ${\rm
AT}\Omega$. In this paper, we will show that the following class of ${\rm
C^*}$-algebras in the class $\Omega$ are inherited by simple unital ${\rm
C^*}$-algebras in the class $\rm T\Omega$ $(1)$ the class of real rank zero
${\rm C^*}$-algebras, $(2)$ the class of ${\rm C^*}$-algebras with the radius
of comparison $n$, and $(3)$ the class of $\rm AT\Omega$. As an application,
let $\Omega$ be a class of unital ${\rm C^*}$-algebras which have generalized
tracial rank at most one (or has tracial topological rank zero, or has tracial
topological rank one). Let $A$ be a unital separable simple ${\rm C^*}$-algebra
such that $A\in \rm AT\Omega$, then $A$ has generalized tracial rank at most
one (or has tracial topological rank zero, or has tracial topological rank
one).
