Let X be a topological space
which is second countable, locally compact, and T0. Fell has defined a compact
Hausdorff topology on the collection 𝒞(X)) of closed subsets of X. X may be
identified with a subset of 𝒞(X), and in the first part of this paper, the original
topology on X is related to that induced from 𝒞(X). The main result is a necessary
and sufficient condition for X to be almost strongly separated. In the second part,
these results are applied to the primitive ideaI space Prim (A) of a separable
C∗-algebra A, giving in particular a necessary and sufficient condition for Prim (A)
to be almost separated. Further information concerning ideals in A which are central
as C∗-algebras is obtained.
