On feebly compact topologies on the semilattice $\exp_n\lambda$

TL;DR

This paper investigates feebly compact topologies on semilattices of the form $ ext{exp}_n ext{lambda}$, characterizing when such topologies are compact, countably compact, or feebly compact, and constructing examples with specific properties.

Contribution

It provides a complete description of compact semilattice $T_1$-topologies on $ ext{exp}_n ext{lambda}$ and establishes equivalences among various compactness conditions for these semilattices.

Findings

All compact $T_1$-topologies on $ ext{exp}_n ext{lambda}$ are described.

Equivalence of compactness, countable compactness, and feebly compactness for $ ext{exp}_n ext{lambda}$ topologies.

Construction of a non-semiregular, countably pracompact topology with discontinuous operation.

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 $\tau_{\operatorname{\textsf{fc}}}^2$ such that $\left(\exp_2\lambda,\tau_{\operatorname{\textsf{fc}}}^2\right)$ 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.