On feebly compact topologies on the semilattice ${\exp }_n\lambda$

Abstract

We study feebly compact topologies $\tau$ on the semilattice $\left({\exp }_n\lambda ,\cap \right)$ such that $\left({\exp }_n\lambda ,\tau \right)$ is a semitopological semilattice. All compact semilattice $T_1$-topologies on ${\exp }_n\lambda$ are described. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ for a $T_1$-topology $\tau$ on ${\exp }_n\lambda$ the following conditions are equivalent: $(i)$ $\left({\exp }_n\lambda ,\tau \right)$ is a compact topological semilattice; $(ii)$ $\left({\exp }_n\lambda ,\tau \right)$ is a countably compact topological semilattice; $(iii)$ $\left({\exp }_n\lambda ,\tau \right)$ is a feebly compact topological semilattice; $(iv)$ $\left({\exp }_n\lambda ,\tau \right)$ is a compact semitopological semilattice; $(v)$ $\left({\exp }_n\lambda ,\tau \right)$ is a countably compact semitopological semilattice. We construct a countably pracompact $H$-closed quasiregular non-semiregular topology ${\tau }_{\mathrm{fc}}^2$ such that $\left({\exp }_2\lambda ,{\tau }_{\mathrm{fc}}^2\right)$ is a semitopological semilattice with discontinuous semilattice operation and prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $\lambda$ every $T_1$-semiregular feebly compact semitopological semilattice ${\exp }_n\lambda$ is a compact topological semilattice.

References in corpus (1)

• Semigroup Closures of Finite Rank Symmetric Inverse Semigroups