Vietoris topology Let F be a closed subset and U_1,\ldots,U_n be a finite collection of open subsets of
X. Define V(F,U_1,\ldots,U_n) = \{ Y \subseteq X \text{ closed} \mid Y \cap F = \emptyset \text{ and } Y\cap U_i \neq \emptyset \text{ for every }1\leq i\leq n \}. These sets form a basis for a topology on
CL(X), called the Vietoris or finite topology, named for
Leopold Vietoris.
Fell topology A variant on the Vietoris topology is to allow only the sets V(C,U_1,\ldots,U_n) where
C is a compact subset of
X and U_1,\ldots,U_n a finite collection of open subsets. This is again a base for a topology on
CL(X) called the Fell topology or the
H-topology. Note, though, that the canonical map i:x \mapsto \overline{\{x\}} is a homeomorphism onto its image if and only if
X is Hausdorff, so for non-Hausdorff
X, the Fell topology is not a hypertopology in the sense of this article. The Vietoris and Fell topologies coincide if
X is a compact space, but have quite different properties if not. For instance, the Fell topology is always compact and it is compact Hausdorff whenever if
X is locally compact. On the other hand, the Vietoris topology is compact if and only if
X is compact and Hausdorff if and only if
X is
regular.
Other constructions The
Hausdorff distance on the closed subsets of a bounded metric space
X induces a topology on
CL(X). If
X is a compact metric space, this agrees with the Vietoris and Fell topologies. The
Chabauty topology on the closed subsets of a locally compact group coincides with the Fell topology. ==See also==