Summary: Following the idea of \textit{J. van Mill} et al. [Topology Appl. 154, No. 10, 2127--2134 (2007; Zbl 1131.54022)], we define a space \(X\) to be neighborhood assignment \(\mathscr{P}\) (briefly NA-\(\mathscr{P}\), where \(\mathscr{P}\) is a class of topological spaces) if for any neighbourhood assignment \(\{O_x : x \in X\}\) there is a subspace \(Y \subset X\) such that \(Y\) has the property \(\mathscr{P}\) and \(\bigcup_{x \in Y} O_x=X\). In this paper we investigate the classes of NA-\(\mathscr{P}\) spaces and compare them with the star covering properties for some class of topological spaces \(\mathscr{P}\).