A note 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 $\leqslant n$. It is proved that such $T_1$-topology is sequentially pracompact if and only if it is feebly compact. Also, we show that every shift-continuous feebly $\omega$-bounded $T_1$-topology on $\mathscr{I}_\lambda^n$ is compact.