The notions of networks and k-networks for topological spaces have played an important role in general topology. Pytkeev networks, strict Pytkeev networks and cn-networks for topological spaces are defined by T. Banakh, and S. Gabriyelyan and J. Kąkol, respectively. In this paper, we discuss the relationship among certain Pytkeev networks, strict Pytkeev networks, cn-networks and k-networks in a topological space, and detect their operational properties. It is proved that every point-countable Pytkeev network for a topological space is a quasi-k-network, and every topological space with a point-countable cn-network is a meta-Lindeï of D-space, which give an affirmative answer to the following problem [25, 29]: Is every Fréchet-Urysohn space with a point-countable cs-network a meta-Lindeï of space? Some mapping theorems on the spaces with certain Pytkeev networks are established and it is showed that (strict) Pytkeev networks are preserved by closed mappings and finite-to-one pseudo-open mappings, and cn-networks are preserved by pseudo-open mappings, in particular, spaces with a point-countable Pytkeev network are preserved by closed mappings.