  Let $\bar{I}$ denote the integral closure of an ideal in a Noetherian ring
$R$. The main result of this paper asserts that $R$ is locally quasi-unmixed if
and only if, the topologies defined by $\overline{I^n}$ and $I^{\langle
n\rangle}$, $\ n\geq 1$, are equivalent. In addition, some results about the
behavior of linearly equivalent topologies of ideals under various ring
homomorphisms are included.
