On feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

Abstract

We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup $\mathscr{I}_λ^n$ of finite transformations of the rank $\leqslant n$. For any positive integer $n\geqslant2$ and any infinite cardinal $λ$ a Hausdorff countably pracompact non-compact shift-continuous topology on $\mathscr{I}_λ^n$ is constructed. We show that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ for a $T_1$-topology $τ$ on $\mathscr{I}_λ^n$ the following conditions are equivalent: $(i)$ $τ$ is countably pracompact; $(ii)$ $τ$ is feebly compact; $(iii)$ $τ$ is $d$-feebly compact; $(iv)$ $\left(\mathscr{I}_λ^n,τ\right)$ is H-closed; $(v)$ $\left(\mathscr{I}_λ^n,τ\right)$ is $\mathbb{N}_{\mathfrak{d}}$-compact for the discrete countable space $\mathbb{N}_{\mathfrak{d}}$; $(vi)$ $\left(\mathscr{I}_λ^n,τ\right)$ is $\mathbb{R}$-compact; $(vii)$ $\left(\mathscr{I}_λ^n,τ\right)$ is infra H-closed. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ every shift-continuous semiregular feebly compact $T_1$-topology $τ$ on $\mathscr{I}_λ^n$ is compact.