On feebly compact topologies on the semilattice \(\exp_{n_{\lambda}}\) (Q2407959)

A topological space \(X\) is called feebly compact if each locally finite open cover of \(X\) is finite. The semilattice announced in the title is \(\exp_n \lambda = \{A\subset \lambda : |A|\leq n\}\) with the operation \(\cap\), where \(\lambda\) is a non-zero cardinal, \(n\in \mathbb{N}\). The semilattice \(\exp_n \lambda\) is isomorphic to the band of symmetric inverse semigroups of finite transformations of rank \(\leq n\), \(n\in \mathbb{N}\). In analogy with [\textit{O. Gutik} et al., Mat. Stud. 32, No. 2, 115--131 (2009; Zbl 1224.22004)], the authors construct on \(\exp_n \lambda\) a unique topology \(\tau_c^n\) which makes \(\exp_n \lambda\) into a compact semilattice with the \(T_1\)-topology (the family \[ \{ \{\uparrow x \backslash (\uparrow x_1 \cup \cdots \cup \uparrow x_j): x_1,\ldots,x_j \in \uparrow x \backslash \{x\}\} \mid x\in \exp_c^n\} \] forms a neighbourhood system for \((\exp_n \lambda, \tau_c^n)\)). The constructed topology makes \(\exp_n \lambda\) into a countably compact semilattice, feebly compact semilattice, countably compact semitopological semilattice and countable compact semitopological semilattice. Further, the authors investigate conditions under which the semitopological semilattice \(\exp_n \lambda\) is a topological semilattice.