Concerning some results of Pettis (Q619608)

Let \(X\) be a topological space. A grill on \(X\) is a non-empty collection \(\mathcal{G}\) of subsets of \(X\) such that (i) if \(A\in \mathcal{G}\) and \(A\subseteq B\subseteq X\), then \(B\in \mathcal{G}\), and (ii) if \(A\subseteq B\subseteq X\) and \(A\cup B\in \mathcal{G}\), then \(A\in \mathcal{G}\) or \(B\in \mathcal{G}\). Like nets and filters, grills can be used in certain topological spaces. In the paper under review, the authors generalize some results of \textit{B. J. Pettis} [Proc. Am. Math. Soc. 2, 166--171 (1951; Zbl 0043.05502)] in terms of grills, an important concept introduced by \textit{G. Choquet} [C. R. Acad. Sci., Paris 224, 171--173 (1947; Zbl 0029.07602)]. Specifically in a second category topological space, they show that any non-empty open set in a second category topological group is of second category. The paper is fundamentally self-contained.