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