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

Анотація

We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup ${\mathcal{I}}_{\lambda }^n$ of finite transformations of the rank $\le n$. For any positive integer $n\ge 2$ and any infinite cardinal $\lambda$ a Hausdorff countably pracompact non-compact shift-continuous topology on ${\mathcal{I}}_{\lambda }^n$ is constructed. We show that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ for a $T_1$-topology $\tau$ on ${\mathcal{I}}_{\lambda }^n$ the following conditions are equivalent: (i) $\tau$ is countably pracompact; (ii) $\tau$ is feebly compact; (iii) $\tau$ is d-feebly compact; (iv) $({\mathcal{I}}_{\lambda }^n,\tau )$ is H-closed; (v) $({\mathcal{I}}_{\lambda }^n,\tau )$ is ${\mathrm{\mathbb{N}}}_{\mathfrak{d}}$-compact for the discrete countable space ${\mathrm{\mathbb{N}}}_{\mathfrak{d}}$; (vi) $({\mathcal{I}}_{\lambda }^n,\tau )$ is $\mathrm{ℝ}$-compact; (vii) $({\mathcal{I}}_{\lambda }^n,\tau )$ 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 ${\mathcal{I}}_{\lambda }^n$ is compact.