On spaces defined by Pytkeev networks

Abstract

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, S. Gabriyelyan and J. K\c{a}kol, 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-Lindel\"{o}f D-space, which give an affirmative answer to the following question \cite{Ls15, LZ15}: Is every Fr\'{e}chet-Urysohn space with a point-countable $cs'$-network a meta-Lindel\"{o}f 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.