On feebly compact topologies on the semigroup BωF1

Анотація

We study the Gutik-Mykhalenych semigroup ${\mathit{B}}_{\mathbf{\omega }}^{{\mathcal{F}}_{\mathbf{1}}}$ in the case when the family ${\mathcal{F}}_1$ consists of the empty set and all singleton in $\omega$. We show that ${\mathit{B}}_{\mathbf{\omega }}^{{\mathcal{F}}_{\mathbf{1}}}$ is isomorphic to  subsemigroup ${\mathcal{B}}_{\omega }^{\nearrow }\left({\omega }_{min}\right)$ of the Brandt $\omega$-extension of the semilattice $\left(\omega ,min\right)$ and describe all shift-continuous feebly compact $T_1$-topologies on the semigroup ${\mathcal{B}}_{\omega }^{\nearrow }\left({\omega }_{min}\right)$. In particular, we prove that every shift-continuous feebly compact $T_1$-topology τ on ${\mathit{B}}_{\mathbf{\omega }}^{{\mathcal{F}}_{\mathbf{1}}}$ is compact and moreover in this case  the space $\left(B_{\omega }^{{\mathcal{F}}_1},\tau \right)$ is homeomorphic to the one-point Alexandroff compactification of the discrete countable space $\mathfrak{D}\left(\omega \right)$.