We prove the following theorems: There exists an $\omega$-covering with the property $s_0$. Under $\mathrm{c}ov\left(𝒩\right)=$ there exists $X$ such that ${\forall }_{B\in ℬor}[B\cap X$ is not an $\omega$-covering or $X\setminus B$ is not an $\omega$-covering]. Also we characterize the property of being an $\omega$-covering.