Relative star normal type (Q818416)

There are many different relative topological properties based on the concepts of regularity, normality, paracompactness, and metrizability. The paper under review explores quite a number of them, answering several questions in \textit{A. Arkhangel'skiĭ} and \textit{I. Gordienko} [Commentat. Math. Univ. Carol. 37 (4), 757--774 (1996; Zbl 0886.54001)] and in \textit{A. Arkhangel'skiĭ} [Sci. Math. Japon. 55 (1), 153--201 (2002; Zbl 0994.54024)]. The following examples suggest the range of possibilities. A subset \(Y\) of a space \(X\) is said to be Aull-paracompact in \(X\) (resp. strongly star normal in \(X\)) iff every collection \(\mathcal U\) of open subsets of \(X\) covering \(Y\) has an open partial refinement \(\mathcal V\) covering \(Y\) which is locally finite on \(Y\) (resp. which star refines \(\mathcal U\) at every point of \(\bigcup \mathcal V\)). A subset \(Y\) of \(X\) is 1-paracompact in \(X\) (resp. 2-paracompact in \(X\)) iff every open cover of \(X\) has an open refinement (resp. a partial open refinement covering \(Y\)) which is locally finite on \(Y\). 3-paracompactness weakens 2-paracompactness further by only requiring each member of the refinement to meet \(Y\) in a relatively open set. Answering all parts of Question 4 in the Arkhangel'skiĭ-Gordienko paper, the authors show that a subspace \(Y\) is Aull paracompact in \(X\) iff \(Y\) is strongly star normal in \(X\). The immediate corollary that \(Y\) is 2-paracompact in \(X\) whenever \(y\) is strongly star normal in \(X\) answers Question 9 of the Arkhangel'skiĭ-Gordienko paper and Problem 12 of the Arkhangel'skiĭ\ paper. Several other implications between related concepts are shown and several open problems are posed relating these to each other and to the ones defined above. Some examples showing various classes to be distict round out the paper. In the references, the title of the Arkhangel'skiĭ-Gordienko paper should be ''Relative symmetrizability and metrizability.''