How to obtain all fine category density topologies (Q1322624)

In the eighties, W. Wilczyński and his colleagues published a series of papers on Wilczyński's \({\mathcal I}\)-density topology [see \textit{W. Wilczyński}, A generalization of density topology, ibid. 8, 16-20 (1983)] which is, in a sense, analogous to the density topology on the line. One of the main analogies consists of the fact that both topologies arise from some ``lower density operator'' and therefore have similar properties. \textit{L. Zajíček} [ibid. 12, No. 1, 313-326 (1987; Zbl 0641.26004)] showed that the \({\mathcal I}\)-density topology is created by the \(*\)-modification of the so-called porous topology. In addition he showed that if in a Baire topological space the \(*\)-modification is applied to a topology that is \(S\)-related to the original topology, then a category density topology (i.e. topology determined by a category lower density operator) is obtained as a consequence. The main result of the present paper contained in the first part is the assertion that every fine category density topology in a Baire regular topological space can be obtained in this way; i.e. every fine category density topology is obtained as the \(*\)-modification of some (called ``primitive'' in the paper) topology, \(S\)-related to the original topology. Moreover, two ways of obtaining the coarsest and the finest of all topologies primitive to the original category density topology are presented. In the second part of the paper, the results are applied to some well-known fine topologies in \(\mathbb{R}\). A consequence of these applications is the observation that a primitive topology generally is not determined uniquely.