  Modifying the known definition of a Pytkeev network, we introduce a notion of
Pytkeev$^*$ network and prove that a topological space has a countable Pytkeev
network if and only if $X$ is countably tight and has a countable Pykeev$^*$
network. In the paper we establish some stability properties of the class of
topological spaces with the strong Pytkeev$^*$-property.
