Chain conditions and star covering properties (Q1725674)

In what follows, all spaces are assumed to be Hausdorff topological spaces, unless otherwise stated. The following properties are examples of \textit{chain conditions} in General Topology. A space \(X\) is said to have the \textit{countable chain condition} (CCC for short) if every disjoint family of non-empty open sets is countable, and it is said to have the \textit{discrete countable chain condition} (DCCC for short) if every discrete family of non-empty open sets is countable. A space \(X\) is \textit{weakly Lindelöf} if every open cover of \(X\) contains a countable subfamily whose union is dense in \(X\). Weak Lindelöfness is usually regarded as being a chain condition because it is, clearly, an intermediate property between CCC and DCCC. A space \(X\) is said to be \textit{weakly \(\omega_1\)-collectionwise Hausdorff}\, if for any closed discrete set \(D \subseteq X\) with \(|D| = \omega_1\) there exists a set \(E \subseteq D\) such that \(|E| = \omega_1\) and \(E\) has a disjoint open expansion (meaning, there is a disjoint open family \(\{U_x: x \in E\}\) with \(x \in U_x\) for every \(x \in E\)). For any topological property \(\mathcal P\), a space \(X\) is said to be a \textit{star \(\mathcal{P}\) space} if for every open cover \(\mathcal U\) of \(X\) there is a subspace \(Y \subseteq X\) such that \(Y\) satisfies \(\mathcal P\) and \(\text{St}(Y, \mathcal U) := \bigcup\{U \in \mathcal U: U \cap Y \neq \emptyset\} = X\). The subspace \(Y\) is then called a \textit{star kernel} of the cover \(\mathcal{U}\). In the paper under review, the authors prove a large number of results relating star \(\mathcal P\) properties (for \(\mathcal{P} \in \{\)DCCC, weakly Lindelöf, CCC\(\}\)) with chain conditions. Typical results of the paper are as follows: \((i)\) a space \(X\) is DCCC if, and only if, it is star DCCC; \((ii)\) a perfect space \(X\) is CCC if, and only if, it is star CCC; \((iii)\) a perfectly normal space \(X\) is weakly Lindelöf if, and only if, it is star weakly Lindelöf; \((iv)\) a Tychonoff star compact space may not be star CCC; \((v)\) if \(X\) is a star CCC first countable space, then the supremum of the sizes of its closed discrete subspaces is not larger than \(2^{\aleph_0}\); and \((vi)\) if \(X\) is a weakly \(\omega_1\)-collectionwise Hausdorff star CCC space, then all of its closed discrete subspaces are at most countable.