Mapping Spaces and cs-networks
In this paper the space of maps from an ℵ0-space to a space Y is studied by means of convergent sequence-networks. The notion of a cs − σ-space, a simultaneous generalization of metric spaces and ℵ0-spaces, is defined, and it is shown that if Y is a (paracompact) cs−σ-space then the mapping space from X to Y is a (paracompact) cs−σ-space when equipped with either the compact-open or the cs-open topology. It is proved that the compact sets are the same in the two topologies. The class of cs − σ-spaces and the class of ℵ-spaces introduced by 0 ‘Meara are shown to be identical in the presence of paracompactness.