# On feebly compact shift-continuous topologies on the semilattice $\exp_n\lambda$

## TL;DR

This paper investigates feebly compact shift-continuous topologies on a specific semilattice, establishing their equivalence with several compactness conditions and $H$-closedness in the context of $T_1$-topologies.

## Contribution

It characterizes the equivalence of various compactness properties and $H$-closedness for shift-continuous $T_1$-topologies on the semilattice $ ext{exp}_n ext{lambda}$.

## Key findings

- Feebly compact topologies are equivalent to countably pracompact and $d$-feebly compact topologies.
- Such topologies are also $H$-closed spaces.
- The results unify several compactness conditions under shift-continuous $T_1$-topologies.

## 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 and prove that for any shift-continuous $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$~$\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$ $\left(\exp_n\lambda,\tau\right)$ is an $H$-closed space.