Let Φ be a system of ideals on a commutative Noetherian ring R, and let S be a multiplicatively closed subset of R. The first result shows that the topologies defined by ${I_a}_{I\in \Phi }$ and ${S\left(I_a\right)}_{I\in \Phi }$ are equivalent if and only if S is disjoint from the quintasymptotic primes of Φ. Also, by using the generalized Lichtenbaum-Hartshorne vanishing theorem we show that, if (R,) is a d-dimensional local quasi-unmixed ring, then $H_{\Phi }^d\left(R\right)$, the dth local cohomology module of R with respect to Φ, vanishes if and only if there exists...