If (X;d) is a metric space and Cl(X) is the family of closed subsets of In this paper we compare the locally finite and Vietoris topology on some classes of nonempty closed subset of a metric space. We find the better class with this property. Then we obtain another results in the problem of comparison of hypertopologies. So consider (X;d) a metric space. Denote by P(X) the family of all nonempty subsets of X. If A ⊂ P(X); we define some topologies on A which extend the initial topology (see, for e.g., (3), (4), (7), (8), (12)). This means that, if we restrict these topologies to the family S(X) of single- ton subsets of X; the topology of induced subspace agrees with the initial topology on X: These topologies are called hypertopologies or hyperspacial topologies. In this category one includes some well-known topologies, like Hausdorff topology H , Attouch-Wets topology AW , Vietoris and bounded- Vietoris topology ( v and bV respectively), proximal and bounded-proximal topology ( P and bP respectively), locally finite topology lf and Wijsman topology W.