ON FELL TOPOLOGY AND THE CONVERGENCE TOPO LOGY

In an unpublished Note, D. H. Fremlin proved the following surprising result: let X be a metrizable topological space, then Fell's topology and the convergence topology (on the set of closed subsets of X) coinc ide if and only if the space X has at most one point with no compact n eighbourhood. In this Note, we compare Fell's topology and the converg ence topology in a more general frame than that of metrizable spaces. In particular, we obtain that, if X is a regular, first countable, loc ally paracompact space, then Fell's topology and the convergence topol ogy coincide if and only if the space X has at most one point with no compact neighbourhood.