This paper deals with certain maximal structures viewed as sets of points that can be endowed with various topologies. Given a \(\pi\)-system \(\mathcal L\) on a set \(E\) (that is, a nonempty family of subsets of \(E\), closed under finite intersections, and containing both \(E\) and \(\varnothing\)), the author considers the set of all maximal linked subfamilies in \(\mathcal L\), as well as the set of all maximal filters of elements in \(\mathcal L\), denoted by \(\langle\mathcal L-\mathrm{link}\rangle_0[E]\) and \(\mathbb F_0^*(\mathcal L)\), respectively (it is clear that the latter is a subset of the former). One can endow the set \(\langle\mathcal L-\mathrm{link}\rangle_0[E]\) with two topologies: one is the topology \(\mathbb T_0\langle E|\mathcal L\rangle\), defined by declaring all sets of the form \[ \{\mathcal E\in\langle\mathcal L-\mathrm{link}\rangle_0[E]\mid(\exists\Sigma\in\mathcal E)(\Sigma\subseteq\Lambda)\}, \] where \(\Lambda\) can be any complement of an element of \(\mathcal L\), to be subbasic open sets. This is said to be a Wallman-type topology. The other topology on the same space that the author considers is defined by declaring all sets of the form \[ \{\mathcal E\in\langle\mathcal L-\mathrm{link}\rangle_0[E]\mid L\in\mathcal E\}, \] with \(L\in\mathcal L\), to be subbasic open sets, and this topology is clearly reminiscent of the Stone topology; it is denoted by \(\mathbb T_*\langle E|\mathcal L\rangle\).
The author also defines two topologies, denoted by \(\mathbf T_{\mathcal L}^0\langle E\rangle\) and \(\mathbf T_{\mathcal L}^*[E]\), on the set \(\mathbb F_0^*(\mathcal L)\). Regardless of how these topologies are defined, they are eventually proved to be the restrictions to \(\mathbb F_0^*(\mathcal L)\) of \(\mathbb T_0\langle E|\mathcal L\rangle\) and \(\mathbb T_*\langle E|\mathcal L\rangle\), respectively. Some results proved in the paper are that \(\mathbb T_0\langle E|\mathcal L\rangle\) is a \(T_1\) supercompact topology, and \(\mathbb T_*\langle E|\mathcal L\rangle\) is Hausdorff (\(T_2\)) and zero-dimensional – with \(\mathbf T_{\mathcal L}^*[E]\) furthermore compact under appropriate assumptions about \(\mathcal L\) – and that \(\mathbb T_0\langle E|\mathcal L\rangle\subseteq\mathbb T_*\langle E|\mathcal L\rangle\) (and consequently, \(\mathbf T_{\mathcal L}^0\langle E\rangle\subseteq\mathbf T_{\mathcal L}^*[E]\) as well); moreover, in the case when \(\mathcal L\) is a lattice (that is, closed under unions in addition to a \(\pi\)-system), then we have equalities rather than inclusions.
The paper itself is hard to read. The mathematical ideas are not outrageously complicated, but it is extremely easy to get lost in notation (the reader of this review has already seen some nuggets of confusing notation in the previous paragraphs: simply changing from square brackets to parenthesis, turning a subscript into a superscript, or switching between varieties of boldface letters, already means that one is referring to a completely different object). There is also a weird remark that in the paper it is assumed that “positive integers are not sets”, so as to avoid some ambiguity that I cannot quite see.