A collection \({\mathcal U}\) of subsets of a topological space \(X\) is said to be \(*\)-open if for each \(x\in X\), \(\text{St}(x,U)\) is open. Replacing “open” by “\(*\)-open” the authors create weak versions of some known concepts. For example, a space \(X\) has weak \(G_ \delta\)-diagonal if there is a sequence \(\{{\mathcal U}_ n\}\) of \(*\)-open covers of \(X\) such that \(\bigcap_ n \text{St}(x,U_ n)=\{x\}\) for each \(x\in X\). This enables them to get generalizations of two cardinal inequalities due to D. K. Burke and R.E. Hodel [Proc. Am. Math. Soc. 58, 363-368 (1976; Zbl 0335.54005)] and J. Ginsburg and R. Grant Wood [ibid. 64, 357-360 (1977; Zbl 0398.54002)].